# Simple Mechanisms for Representing, Indexing and Manipulating Concepts

Yuanzhi Li Carnegie Mellon University<EMAIL_ADDRESS>Raghu Meka
Google Research<EMAIL_ADDRESS>
Rina Panigrahy Google Research<EMAIL_ADDRESS>Kulin Shah UT Austin
<EMAIL_ADDRESS>Work done during an internship at Google Research.

###### Abstract

Deep networks typically learn concepts via classifiers, which involves setting
up a model and training it via gradient descent to fit the concept-labeled
data. We will argue instead that learning a concept could be done by looking
at its moment statistics matrix to generate a concrete representation or
signature of that concept. These signatures can be used to discover structure
across the set of concepts and could recursively produce higher-level concepts
by learning this structure from those signatures. When the concepts are
‘intersected’, signatures of the concepts can be used to find a common theme
across a number of related ‘intersected’ concepts. This process could be used
to keep a dictionary of concepts so that inputs could correctly identify and
be routed to the set of concepts involved in the (latent) generation of the
input.

##  1 Introduction

Deep networks typically learn concepts via classifiers [QYSG17, AZL20, LKL14,
ZWLD16, JT20, Guo21], which involves setting up a model and training it via
gradient descent to fit the concept-labeled data. However, this does not
provide a general understanding of what a concept should be and how they may
be discovered and represented. In this work, we attempt to provide a formalism
of concepts that addresses these issues. Intuitively, we understand a concept
to be a collection of objects with a shared set of properties. In the paper,
we consider such properties as defined by a polynomial manifold, i.e., the
zero set of polynomial equations.

While there has been work on learning manifolds [Ize12, CLGPC21, BBR+18,
PGG18, WYN+18, LZ08, BC20, BTI13, HLW+22, C+08, MF11, ZLC+18, LPCE13, TKR08],
our goal here is to find simple ways to automatically group points into
collections of manifolds and obtain simple representations of these manifolds.
We show that by simply taking the moment statistics corresponding to such
points and looking at the null-space of these statistics (arranged as a
matrix), we can identify the manifold. In other words, these elementary
statistics can serve as a _signature_ of the concept. Thus, we are learning a
concept by tracking its elementary statistics. This signature can serve as a
classifier and as a representation or signature of that concept.

Importantly, these signatures can be used to discover structure across the set
of concepts and could recursively produce higher-level concepts by learning
this structure from those signatures. Concepts may be “intersected” to find a
common theme across a number of related concepts. For a simple illustration of
this abstraction, see Figure 1.

We also map the process of concept discovery to the transformer architecture
[VSP+17] at an intuitive level. We view a generalized transformer architecture
with attention blocks that are used to group together inputs that are part of
the same concept. The MLP layers could then be used to compute the moment
statistics and the corresponding null-space leading to the signature of
manifolds. We also augment each layer with a dictionary of already discovered
concept signatures (viewed as a memory table of concepts). These tables can be
looked up to identify known concepts that are present in a given input. This
process is repeated iteratively producing a hierarchy of concepts.

###  1.1 Our contribution

Our main results can be summarized as follows:

######  Proposition 1.1.

(Informal summary of results) Given a collection of concepts where each
concept corresponds to a (latent) low dimensional, low curvature manifold,
there is a simple method to compute concept signatures such that

  * 1

The concept signature is obtained by taking moment statistics of a few points
within the manifold, followed by applying a power iteration-like transform
that results in a signature that is invariant across different regions in the
manifold. That is, regardless of which region in the manifold the points may
be distributed from, the resulting concept signature will be the same as long
as the distribution has a “well-conditioned” structure (Proposition 2.1).

  * 2

A simple attention mechanism may be used to identify a set of points from the
same manifold (Proposition 3.1). This simply involves looking at the inner
product of the two points after a simple kernel transform. Membership in a
manifold is also a simple inner product check between the concept signature
and this kernel transform representation of the point (Theorem 2.1).

  * 3

If there is structure across a collection of concepts, then this structure is
also present in the concept signatures. For example, a concept may be the
intersection of two other concepts; that is the manifold is the intersection
of the two corresponding manifolds; then this structure can be easily inferred
from the concept signatures. If several similar concepts share a common
underlying (latent) concept, the common concept signature can be easily
obtained from a few of those related concepts (Proposition 3.2).

If there is a dictionary of atomic concepts and all concepts are unions of
these atomic concepts, then the signature of the atomic concepts can be
inferred (Lemma 3.2).

  * 4

Higher level concepts may be obtained when concept signatures at a lower level
lie in a manifold. For example, a single rectangle is a concept that includes
the concept of the four line segments. Each line segment has a concept
signature and these four signatures line a subspace.

The general concept of a circle/rectangle is obtained by taking signatures of
several circles/rectangles that all lie on a single abstract manifold and
computing the concept signature of that manifold (see Lemma 4.1 for circle).

The video of an object moving in a specific motion pattern – for example, a
car turning right will result in a concept signature that will be related to
that of any other object moving in the same pattern. By intersecting the two
concept signatures we obtain the concept of ”moving right” (see Lemma 4.4).

Similarly rotated copies of an image result in concept signatures that lie in
a manifold. These manifolds corresponding to rotations of different images
share a common rotation manifold that captures the concept of rotating any
image. The concept signature of rotation can be used to take the signature of
an image and convert it into a rotation invariant signature which is the
signature of the manifold that includes all rotations of that object (Lemma
4.3).

  * 5

There are simple instantiations of a transformer network with mostly random
set of parameters and a few learned projections that can be used to group
together and compute concept signatures of latent manifolds present in the
input data (see Section 6).

Our analysis points towards an architecture that is very similar to the
transformer but in addition also has a dictionary of concepts signatures at
each layer that are produced over time in a system that is receiving a
continuous stream of inputs; this set of signatures is looked up as new inputs
arrive into the system.

(a) Several lines on a 2-d plane, each with a separate signature.

(b) A rectangle with 4 segments whose signatures are related. The signatures
of these four signatures form a signature of the full rectangle. Similarly for
the person.

(c) Several concentric circles of varying radius: the signatures of the
individual circles all lie on a manifold that is the signature of the concept
of a circle.

(d) Two groups each traveling along a different trajectory. The trajectory of
each object produces a signature; all objects in one group have related
signatures that lie in a manifold.

Figure 1:  Examples and properties of the signatures.

##  2 Concepts as subspace signatures

In this section, we define our main notion of a _signature_ for a set of
points that is meant to capture the notion of concepts. We then show that when
the underlying points lie on an _algebraic manifold_ (see definition below),
the signature we define satisfies the conceptual requirements outlined in the
introduction.

We will use the following notational conventions: For two tensors $A,B$ of the
same dimension, we use $A.B$ to denote $\langle A,B\rangle\in\mathbb{R}$. For
any vector $x$, $x^{.l}$ denotes the $l$’th tensor power of $x$.

######  Definition 2.1.

Given a feature mapping $\phi:\mathbb{R}^{d}\to\mathbb{R}^{m}$ and a
collection of points $x_{1},x_{2},\cdots,x_{N}\in\mathbb{R}^{d}$, we define
the kernel signature of $X=(x_{1},\cdots,x_{N})$ as:

$M=M(X)=\frac{1}{N}\sum_{i\in[N]}\phi(x_{i})\phi(x_{i})^{\top}\in\mathbb{R}^{m\times
m}$

For a single point, we also define $S(x)=\phi(x)\phi(x)^{\top}$.

Given $U\Sigma U^{\top}$ be the Singular Value Decomposition (SVD) of $M$,
where
$\Sigma_{1}\geq\Sigma_{2}\geq\Sigma_{k}>0,\Sigma_{k+1},\cdots,\Sigma_{m}=0$,
we define the null-space signature as $T(X)=U_{k:m}U_{k:m}^{\top}$. We define
the $\epsilon$-approximate null-space signature as
$T_{\epsilon}(X)=U_{k^{\prime}:m}U_{k^{\prime}:m}^{\top}$, where
$\Sigma_{k^{\prime}}\leq\epsilon$ and $\Sigma_{k^{\prime}-1}>\epsilon$.

######  Definition 2.2 (Signature).

A signature of a set of points is defined as $(M(X),T(X),T_{\epsilon}(X))$ for
some prescribed $\epsilon$.111We keep $T_{\epsilon}(X)$ so in the later
section, we will discuss if $X$ only approximately lies on the manifold
$\mathcal{M}$, but not exactly.

######  Definition 2.3 (Algebraic Manifold).

A set $\mathcal{M}\subseteq\mathbb{R}^{d}$ is an algebraic manifold if
$\mathcal{M}=\\{x:P(x)=0\\}$ where
$P:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d-k}$ is a set of $d-k$ polynomial
maps. The manifold in this case is $k$-dimensional and we say the degree of
the manifold is the max degree of the polynomials $P_{i}$.

Throughout, we adopt the convention that the defining polynomial equations
satisfy the normalization $E_{y\in\mathbb{S}^{d-1}}[P_{i}(y)^{2}]=1$ for
$i\in[d-k]$.

Given a $k$-dimensional manifold as above, if it exists, we call
$G:\mathbb{R}^{k}\rightarrow\mathbb{R}^{d}$ a generative representation of
$\mathcal{M}$ if the following holds: for all $x\in\mathbb{R}^{d}$,
$x\in\mathcal{M}$ if and only if there exists $z\in\mathbb{R}^{k}$ such that
$x=G(z)$.

More generally an analytic manifold is obtained when the functions $G$ are
analytic (possibly in a bounded region).

We will look at manifolds where the dimensionality $k$ and the degree are
small constants – as a simple example a circle is a $1$-d manifold of degree
$2$. We will argue later how rich concepts can be formed by combining several
such simple concepts hierarchically.

######  Definition 2.4 (Non-degenerate distribution).

We will say that a distribution of points from manifold is non-degenerate if
for any subset of features in $\phi(x)$ that are linearly independent over the
entire manifold are also linearly independent over the support of the
distribution. For example, if $\phi$ a polynomial kernel, and the manifold is
analytic, a distribution supported on a (even a small) ball on the manifold is
non-degenerate (this is because if a polynomial is identically zero within a
ball then it is identically zero over the entire analytic manifold)

######  Proposition 2.1.

[Moments null space is zero polynomials on the manifold] Let
$X=\\{x_{1},\ldots,x_{n}\\}$ be random points drawn from a $k$-dimensional
algebraic manifold $\mathcal{M}$ of degree $\ell$.

Under suitable non-degeneracy conditions on $\mathcal{M}$, for sufficiently
large $n$, if we compute the signature $T(X)$ with $\phi$ being the degree
$\ell$-polynomial feature mapping (i.e., $\phi(x)$ contains all monomials of
$x$ of up to degree $\ell$), then the signature $T(X)$ as in Definition 2.1
uniquely identifies the manifold $\mathcal{M}$. Membership check: A point
$x\in\mathcal{M}$ iff $S(x).T(X)=0$, otherwise $S(x).T(X)>0$.

###### Proof.

Let $p:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a polynomial that vanishes on
the manifold $\mathcal{M}$. Then, for any point $x$, $p(x)=\langle
p,\phi(x)\rangle$ (here we view $p$ as the coefficient vector of the
polynomial $p$). Thus, if $p(x)=0$ for all $x\in\mathcal{M}$, then $p$ is in
the null-space of $M(X)$ as $p^{T}\phi(x_{i})\phi(x_{i})^{T}p=0$ for all
$x_{i}$.

For large enough $n$, assuming $\mathcal{M}$ is non-degenerate, any polynomial
that is non-zero on a ball of the manifold will not be in the null-space of
$M(X)$ (with high probability).

Thus, a point $x$ is in the null-space if and only if
$\phi(x)^{T}T(X)\phi(x)=0$, i.e., $S(x).T(X)$ is $0$. ∎

For constant dimensional manifolds with a generative representation with
constant degree polynomials, the sample-size required in the statement above
can be reduced to a constant size (independent of the ambient dimension). The
signature size can also be reduced to a constant size.

######  Proposition 2.2.

[Invariant concept signatures] Suppose $X$ lies in a $k$-dimensional
polynomial manifold with a degree $r$ generative representation. Suppose
$\phi(x)$ is a degree $s$ monomial feature mapping, then $T(X)$ is uniquely
defined by the signature as long as $X$ is non-degenerate and
$N=\Omega\left({k^{r}+s\choose s}\right)$.

###### Proof.

We can write all the monomials of $x$ as a vector, whose coordinates are the
coefficients in terms of $t\in\mathbb{R}^{k}$, where $t$ are the generative
variables. We know that for degree $q$ monomials, there are at least
${d\choose q}$ such terms. On the other hand, we can write the degree $rq$
polynomials $p$ of $t$ as a vector in dimension at most $k^{rq}$. Taking
$s=1$, we know that the manifold lies in a $k^{r}$ dimensional subspace $U$.

We now need to prove that $T(X)$ is uniquely defined. To see this, let us
consider a $w$ such that $\langle w,\phi(x_{i})\rangle=0$ for all $i\in[N]$.
We need to show that $\langle w,\phi(x)\rangle=0$ for all $x\in Y$. Note that
by our previous conclusion,

$\phi(x)=\phi(UU^{\top}x)$

Therefore we can restrict ourself to polynomials $g(z)=\phi(Uz)$, where
$z\in\mathbb{R}^{k^{r}}$.

Now, for each polynomial $g(z)$, we know that $g(U^{\top}p(t))$ is of degree
$rs$. Therefore, for non-degenerate $t_{1},\cdots,t_{N}$ where $N=\Omega(rs)$,
$g(U^{\top}p(t_{i}))=0,\forall i\in[N]\implies g(U^{\top}p(t))=0$. Note that
the VC dimension of $g$ is at most ${k^{r}+s\choose s}$ since $g$ is a
polynomial of degree at most $s$ over input dimension $k^{r}$, so we know that
the sample complexity is $O\left({k^{r}+s\choose s}\right)$. ∎

Now we want to quantitatively show that the signature can be used to do a
membership check if the points lie in a manifold, thus the signature of those
points will precisely describe the manifold.

We also show that we can reduce the sample complexity to
$r^{\tilde{O}(k^{2})}$ by using a random projection to a lower-dimensional
space. The following follows easily from Theorem 1.6 in [Cla08] (the statement
there is about a $k$-dimensional manifold; we can view adding an extra point
as a $(k+1)$-dimensional manifold).

######  Lemma 2.1 (Reducing sample complexity using projection, [Cla08]).

Let $\mathcal{M}\subseteq\mathbb{R}^{d}$ be a connected, compact, orientable,
differentiable manifold M with dimension $k$, constant curvature and bounded
diameter. Then, for any $\epsilon,\delta>0$ and a point $x,\mathbb{R}^{d}$, if
$A:\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}$ is a random linear map for
$m=O((k+\log(1/\delta))/\epsilon^{2})$, with probability at least $1-\delta$,
the following hold:

  * •

For all $y,z\in\mathcal{M}$, $\|Ay-z\|_{2}=(1\pm\epsilon)\|y-z\|_{2}$.

  * •

For all $y\in\mathcal{M}$, $\|Ax-Ay\|=(1\pm\epsilon)\|x-y\|_{2}$.

We use the projection lemma to obtain the following.

######  Theorem 2.1.

[$k$-dim manifold generated by degree $r$ polynomials] A $k$-dim manifold with
generative representation $G$ that are polynomials of degree $r$, can be
represented by a signature $T$ of size $r^{\tilde{O}(k^{2})}$ (independent of
ambient dimension $d$). Such a signature can be computed by projecting the
points to a lower dimensional space and computing the signature as before.
This signature can be used to test the membership of a point $x$ with high
probability by checking the value of $S(x).T$: $S(x).T=0$ if $x$ is on the
manifold. In addition, for any point $y$ on the manifold, if we choose
$z\in\\{x:\|x-y\|_{2}=\Delta\\}$ at random, then $S(z).T\geq f_{k,r}(\Delta)$,
where $f$ is an increasing function in $\Delta$ in a bounded region.

###### Proof.

We first note that if there is a generative representation of a
$k$-dimensional manifold by polynomials of degree $r$, then the degree of the
manifold is at most $\ell=k^{r}$ \- see Theorem 5.1.

Note that any manifold with generative representation of degree $r$ continues
to have a generative representation of same degree after a projection.

Thus, if we use a random projection to project the space to
$m=O((k+\log(1/\delta))/\epsilon^{2})$, then by Lemma 2.1, all distances on
the manifold and the test point $x$ are preserved with probability at least
$1-\delta$.

Let us work in the projected space of dimension $m$. Now, the number of
monomials involved in the monomial map is $\binom{m+\ell}{\ell}$ which is at
most $(m+\ell)^{m}$. Since $\ell$ is at most $r^{k}$, the dimension of the
monomial map is $r^{km}$. Now, note that to approximate $T(X)$, we need a
spectral approximation to $M(X)$. Recall that for any distribution on vectors
in dimension $D$ with bounded covariance, the number of samples needed for the
empirical covariance matrix to approximate the true covariance matrix is
$\tilde{O}(D)$. By applying this to $\phi(x)$ (in the projected space), the
number of samples needed for $M(X)$ to approximate the true moment matrix is
$\tilde{O}(r^{km})$.

Thus, the size of the signature is $\tilde{O}(r^{km})=r^{\tilde{O}(k^{2})}$.

Finally, for the last part, consider the polynomial $Q(x)=S(x).T$ (in the
projected space). By our assumptions, the sum-of-squares of the coefficients
of $Q$ is exactly the dimension of the null-space of $T$. The final part of
the lemma now follows by Lemma C.4. ∎

##  3 Learning Architecture augmented with table of Concept signatures at each
layer

We envision an architecture where a dictionary of signatures
$T_{\ell,1},T_{\ell,2}\ldots,$ are maintained at each layer $l$ (Figure 2).
The concepts at layer $l+1$ are obtained by combining similar concepts at
layer $l$ hierarchically. Using this idea, we propose the following learning
method:

  1. 1.

Given an input vector $x_{t}$ at time $t$.

  2. 2.

For each layer $\ell\in[L]$, we store a sequence of $m$ past signatures
$T_{1,1},T_{1,2},\cdots,T_{1,m}$, where $T_{1,i}=x_{t+1-i}$.

  3. 3.

We compute the attention (as described later) of $T_{\ell}(x_{t})$ and
$T_{\ell,1},T_{\ell,2},\cdots,T_{\ell,m-1}$, obtain the
$T_{\ell,i_{1}},T_{\ell,i_{2}},\cdots,T_{\ell,i_{K}}$ with the highest $K$
attention score, then compute the signature of them to get
$T_{\ell+1}(x_{t})$. We then update $T_{\ell+1,i}=T_{\ell+1,i-1}$ for $i\geq
2$ and $T_{\ell+1,1}=T_{\ell+1}(x_{t})$ to maintain $m$ signatures per layer.

Figure 2: Overview of our learning architecture.

#### The mechanism behind the learning method.

Here, the attention is simply based on $\frac{\langle
x,y\rangle}{\|x\|_{2}\|y\|_{2}}$. We will discuss the mechanism of attention
in the next section, at a high level, our results show that:

  1. 1.

Points lying in a low dimensional manifold are more likely to attend to each
other ( Proposition 3.1), so $T_{\ell+1}(x_{t})$ will more likely be a
signature for the low dimension manifold that is spanned by a subset of
$T_{\ell,i_{1}},T_{\ell,i_{2}},\cdots,T_{\ell,i_{K}}$. Thus, if there is a
subset of signatures in $\\{T_{\ell,2},T_{\ell,3},...,T_{\ell,m}\\}$ that lies
on a low dimension manifold that contains $T_{\ell}(x_{t})$, then
$T_{\ell+1}(x_{t})$ is more likely to be the signature of that manifold.

  2. 2.

Signatures corresponding to the transformation of the same object are more
likely to lie in a low-dimensional manifold.

As an example, if there is a set of $K-1$ points
$x_{i_{1}},x_{i_{2}},\cdots,x_{i_{K-1}}$ and $x_{t}$ that lies in the same
manifold, then $T_{2}(x_{t})$ is more likely to be the signature of that
manifold.

###  3.1 Structure across concepts is present in the signatures

We first show how attention based on distance between points to group points
from the same concept. We also show for linear manifolds (specifically
subspaces) that similar concepts have similar signatures and the signature of
intersection of multiple linear manifolds can be obtained from the individual
signatures of the manifold.

######  Proposition 3.1.

[Attention based on cosine similarity] Given a $k$ dimensional manifold $X$
with constant distortion, for every $\varepsilon>0$, $N\in\mathbb{Z}^{+}$, for
every set of $\Omega(N(\log(k)/\varepsilon)^{k})$ points in
$\\{x\mid\|x\|_{2}\leq 1\\}$, there must be a set $S$ of $N$ points such that
for every $x,x^{\prime}\in S$, $\frac{\langle
x,x^{\prime}\rangle}{\|x\|_{2}\|x^{\prime}\|_{2}}\geq 1-\varepsilon$.

Note that different attention heads could apply attention at different scales
of granularity $\varepsilon$.

###### Proof.

First, we can consider a $k$-dimensional subspace, then the unit ball in the
$k$-dimensional subspace can be covered by $O((\log(k)/\varepsilon)^{k})$ rays
of angles at most $\varepsilon$. By pigeon-hole principle, we complete the
proof. For a general $k$-dimensional manifold with constant distortion, we can
map it back to $k$-dimensional subspace which preserves distance up to a
constant multiplicative factor. ∎

The proposition implies that random points on low-dimensional manifolds are
much closer to each other in general, compared to random points over the
entire space. Thus, as we will show later, points on a single low-dimension
manifold are more likely to attend to each other in an attention layer.

Next we show how two the intersection of two concepts can be learned just from
the signatures of the undelying concepts. We also show that for two linear
manifolds that are subspaces, their signatures have a high dot product if the
subspaces have a higher overlap (i.e., the common subspace is of higher
dimension).

######  Proposition 3.2.

Given two concepts with manifolds $U_{1}$ and $U_{2}$, the intersection of the
two concepts is given by the intersection of the manifolds $U_{1}\cap U_{2}$.
The signature of this intersection can be easily computed as follows. Define
$F(U)=I-T(U)$. Then,

$F(U_{1}\cap U_{2})=(F(U_{1}).F(U_{2}))^{\infty}$

Here by the $\infty$ in the power can be thought of as taking a very high
power.

Further if they are linear manifolds that are subspaces of dimension $k$ then
the similarity between their signatures increases with the dimensionality
$\text{dim}(U_{1}\cap U_{2})$ of their intersection.

  1. 1.

$F(U_{1}).F(U_{2})\geq\text{dim}(U_{1}\cap U_{2})$

Note that
$T(U_{1}).T(U_{2})=F(U_{1}).F(U_{2})+d-\text{dim}(U_{1})-\text{dim}(U_{2}).$

  2. 2.

In contrast two random subspaces $U_{1}$ and $U_{2}$ of dimension $k$ in
$d$-dimensional space have a small dot product if $k^{2}<<d$.

$E_{U_{1},U_{2}}[F(U_{1}).F(U_{2})]=k^{2}/d$

######  Lemma 3.1 (Similar Manifolds).

Consider a manifold specified by a degree $l$ equation written in the form
$c.\Phi(x)=0$ where $c$ is a (normalized) coefficient vector Given two
manifolds $U_{1},U_{2}$ specified by two such coefficient vectors
$c_{1},c_{2}$ $T(U_{1}).T(U_{2})=(c_{1}.c_{2})^{2}$

Next we show that if concepts have a dictionary structure formed from atomic
concepts then such a structure can be discovered easily

######  Lemma 3.2.

[A Dictionary of Concepts] If the concepts have a structural relationship of
the form that there is a set of latent atomic concepts and all concepts are
obtained by taking sparse unions of these atomic subspaces, then this
structure can easily be discovered from the concept signatures.

###### Proof.

This follows from the fact that the signature of the intersection of two
concepts can be obtained from the signatures of the individual concepts
Proposition 3.2. By repeated taking intersections we get the set of atomic
concepts. ∎

###  3.2 Concepts that are Union of a few simpler concepts

######  Remark 3.1.

A stick figure human would consists of about 6 curves drawn on a 2d plane. At
the first level we would get signatures for each of those curves. at the
second level we would get a signature of the curve signatures. This would
become the signature of the entire stick figure. Even though the total degrees
of the freedom here is large, one can easily identify the common invariant
part of the human stick figure concept from just a few examples.

The following Lemma demonstrates how a higher level concept obtained by
combining a few simpler concepts can be represented by the signature of the
signatures of the underlying concepts; for example, a rectangle is obtained by
a union of four line segments and the human stick figure would consist of six
curves.

######  Lemma 3.3 (Moment statistics memorize a small set of points).

For $k$ points, the signature obtained by the $O(k)$th moment statistic
uniquely identifies exactly the set of $k$ points.

###### Proof.

This follows from the fact that there is a polynomial of degree $2k$ whose
zero’s coincide with the set of given $k$ points. This polynomial must be in
the null space of the moments statistic matrix. ∎

##  4 Signature of signatures

As outlined in the introduction, the right notion of signatures and concepts
should allow us to build higher-level concepts from lower-level concepts and
allow us to identify common traits in objects by even simply intersecting
concepts.

As a proof of concept, we show that our idea of building signatures from
statistics can be used to learn simple higher-level concepts from lower-level
concepts. We start with learning the concept of a given circle by looking at
the null-space formed by the points on the circle. Next, by looking at the
signatures of several circles and the null-space of the moments of these
signatures we learn the general concept of a circle. Similarly, we show that
rotated versions of an image lie on a manifold which can be used to obtain a
rotation invariant representation of images.

Further, the idea of intersecting concepts can be used to define a common
direction of motion from several moving objects that share the direction of
motion (i.e., we identify the direction of motion as a common concept for
these different objects).

###  4.1 Signature of a concept of a circle

######  Lemma 4.1.

By setting $S(x)=(\phi(x))^{.2}$, where $\phi(x)=[1;x]^{.2}$, we obtain an
invariant signature of a unit circle $T(x)$. This signature can be obtained
even from random points from an arc of the circle.

###### Proof.

Note that $\phi(x)$ includes all monomials of degree 2 (that is
$\phi(x)=[1,x_{1},x_{2},x_{1}^{2},x_{1}x_{2},x_{2}^{2}]$). The circle
signature is given by $T(X)=I-ww^{\prime}$ where $w=[-1,0,0,1,0,1]$. This
corresponds to the equation of the circle $T(X).S(x)=0$ or $\phi(x).w=0$ or
$x_{1}^{2}+x_{2}^{2}-1=0$. This is because points from the arc can only
satisfy this one equation of degree $2$. Instead of $\phi(x)=[1;x]^{.2}$, if
we had used a higher power $l$, then too any polynomial equation satisfied by
an arc must be satisfied by the circle and must have $x_{1}^{2}+x_{2}^{2}-1$
as a factor. ∎

######  Lemma 4.2.

Individual signatures $T(X)$ of several concentric circles lie in a $1$-d
manifold of degree $2$; the signature of these signature corresponds to the
the concept of a circle (centered at a origin).

###### Proof.

The nullspace of a circle of radius $r$ is given by the equation
$x_{1}^{2}+x_{2}^{2}-r^{2}=0$ which corresponds to the singular vector
$w=[-r^{2},0,0,1,0,1]$. Thus, the signature $C_{r}$ of a circle of radius $r$
is $I-ww^{\top}$ where $w=[-r^{2},0,0,1,0,1]$

Thus $C_{r}=a_{1}+a_{2}r+a_{3}r^{2}$ where $a_{1},a_{2},a_{3}$ are constant
vectors when flattened. Thus we can think of $C_{r}$ as isomorphic to
$[z_{0},z_{1},z_{2}]=[1,r^{2},r^{4}]$. Thus under this isomorphic view $C_{r}$
is the intersection of the polynomial equations: $z_{0}-1=0,z_{2}-z_{1}^{2}=0$

Thus $T(z)=T(C_{r}^{.2})=z^{.2}(I-w_{1}w_{1}^{\prime}-w_{2}w_{2}^{\prime})$
where $w_{1}.z=0$ and $w_{2}.z=0$ capture the above two equations. This is
because if we take $[1,r^{2},r^{4}]$ for a few different values of $r$ any
degree $4$ equation $z^{.2}$ satisfies is satisfied by all such
$[1,r^{2},r^{4}]$. ∎

###  4.2 Rotation invariant signature

In this section, we show how rotating an image produces the signatures on an
analytic manifold which means we can use Theorem 5.2 to obtain a signature of
this manifold.

######  Lemma 4.3.

Let $X$ denote the point cloud of the pixels of image (each pixel can be
viewed as concatenation of position and color coordinates) and let
$X_{\theta}$ denote the rotation of the image under rotation $\theta$. Then,
$S(X_{\theta})$ can be written as a Taylor series in $\theta$ and therefore,
we can obtain the rotation invariant signature of $X$ from the signature of
signatures $\\{S(X_{\theta})\\}_{\theta}$.

See Appendix B for the Taylor series approximation of the rotation and an
additional example of translation invariant signature of $X$.

###  4.3 Signature of a object moving in a specific motion pattern

######  Lemma 4.4.

The signatures of the motion of two objects moving with the same velocity
function is sufficient to obtain the signature of that velocity function. Any
other object with same velocity function will match that signature. Given a
set of points $X_{1},X_{2}$ each moving along separate velocity functions
$v_{1}(t),v_{2}(t)$, will result in a collection of concept signatures for
each point’s trajectory. All the concept signature of point trajectories from
one set will intersect in the corresponding velocity concept signature.

###### Proof.

Different objects moving with same velocity can be viewed as subspaces that
intersect in the common velocity subspace. For e.g. an object moving in 3d
space can be viewed as 1d manifold. And if the velocity is common these are
parallel manifolds. By appending a $1$ coordinate these can be seen as 2d
manifolds that intersect in the common velocity manifold. By Proposition 3.2
two such signatures are sufficient to obtain the signature $F(V)$ of the
velocity manifold. Checking if another object motion $F(O)$ has the same
velocity is also easy: check if $F(O)F(V)==F(V)$.

∎

##  5 Properties of Manifolds

In this section, we study properties of manifolds and show auxiliary results
for polynomials. In particular, we consider the manifold that is implicitly
represented by a generator of $k-$dimensional space and show that there exists
a polynomial $H(x)$ that represents the manifold. We first start with the case
when the generator of the data is low degree polynomial, i.e., $X=G(z)$ where
Generator $G:\mathbb{R}^{k}\to\mathbb{R}^{d}$ is a degree $r$ polynomial. We
also show that polynomials with non-trivial norm are unlikely to be very close
to zero; this is useful for checking membership on manifolds. Due to space
constraints, we only provide theorem statements in this section and provide
proofs in Appendix C.

###  5.1 Explicit polynomial representation from implicit polynomials

######  Theorem 5.1 (Explicit polynomials from implicit polynomials).

Suppose $X\in\mathbb{R}^{d}$ lies in a $k-$dimensional manifold and is
represented as a degree $r$ implicit polynomial by $X=G(z)$ where
$z\in\mathbb{R}^{k}$. Then, the manifold can be written as zero sets of $d-k$
polynomials $H(X)$ and the degree of each of the polynomials is at most
$r^{k}$.

The proof uses the fact that when the data generator equation $X=G(z)$ is
raised to the sufficiently high tensor power $u$ to obtain $X^{.u}=G(z)^{.u}$,
then there will be more different monomials in $X^{.u}$ than $G(z)^{.u}$
because $X$ has more number of variables than $z$. Therefore, one can
eliminate the implicit variables $z$ to obtain a polynomial representation
corresponding to the manifold.

###  5.2 Approximate polynomial representation for analytic manifolds

Next, we show that even if $G(z)$ is not a polynomial but a general analytic
function, then the manifold can be approximated as $d-k$ polynomials of high-
enough degree.

######  Theorem 5.2.

Suppose a $k-$dimensional manifold of the data $X\in[-1,1]^{d}$ is given by
$X=G(z)$ where $G:[-1,1]^{k}\to[-1,1]^{d}$ is an analytic function and
$\|\nabla^{(m)}G(z)\|_{2}\leq 1$ for any $m$. Then, there exists a polynomial
in $X$, $H(X)$ of degree $r$ such that for points on the manifold,
$\|H(X)\|_{\infty}\leq r^{-\frac{r^{(1+1/k)}}{2c^{k}k}}$ for some absolute
constant $c$.

##  6 Connection to Transformer Architecture

Our learning architecture includes attention mechanism that groups from the
same concept together and MLP layer which computes the signature of the
concept and a memory table which keeps track of moment signature seen so far.
Even though we presented the signatures as matrices, we will argue how this
signatures will arise from flattened vectors from random MLP layers with small
amount of trained projection.

In particular, we show that a two-layer neural network with random weights can
obtain $M(X)^{2}$ upto rotation, i.e., there exists a fixed projection of
representation produced by the MLP layers that equals $M(X)^{2}$. One can use
this two-layer neural network as a building block to obtain the higher powers
of $M(X)$ and the null-space of a high enough powers of $M(X)$ defines the
null-space signature of the concept.

We denote the data matrix $\mathbf{X}\in\mathbb{R}^{N\times d}$ where $i^{th}$
row is given by $x_{i}$. We consider the square activation and the following
two-layer neural network with $m$ hidden units whose weights are initialized
using standard Gaussian $\mathcal{N}(0,I)$:

$\displaystyle G_{1,j}(x_{i})$ $\displaystyle=\sigma(x_{i}\cdot
r_{j}),\;\;\;\;\hat{G}_{1,j}=\frac{1}{N}\sum_{i=1}^{N}G_{1,j}(x)\;\;\text{and}\;\;G_{2,j}=\sigma(\hat{G}_{1,j})$

where $\sigma$ is an element-wise square activation. For wide enough MLP
layers, we show that $\hat{G}_{1,j}$ is equivalent to $M(X)$ by a projection
and $G_{2,j}$ is equivalent to $M(X)^{2}$ upto a projection.

######  Theorem 6.1.

There exists a projection of $\hat{G}_{1,j}$ that gives $M(X)$:

$\displaystyle
E_{r\sim\mathcal{N}(0,I)}[r_{j}^{(.2)}\hat{G}_{1,j}]-(d+1)I=M(X).$

Similarly, there exists a projection of the representation generated by first
and second layer such that

$\displaystyle E_{r\sim\mathcal{N}(0,I)}[r_{j}^{\otimes
2}\hat{G}_{1,j}+\beta_{1}r_{j}^{\otimes 2}G_{2,j}]+\beta_{2}I=M(X)^{2}$

where $\beta_{1}$ and $\beta_{2}$ only depend on $d$.

The full proof can be found in Appendix D.

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##  Appendix A Proofs for Section 3

###  A.1 Proof for Lemma 3.1

###### Proof.

If a manifold $U$ is specified by a polynomial equation of degree $l$ and we
use $lth$ moments, there will be exactly one eigenvector in the null space
specified by $c$. Thus $T(U)=cc^{\prime}$. So
$T(U_{1}).T(U_{2})=c_{1}c_{1}^{\prime}c_{2}c_{2}^{\prime}=(c_{1}.c_{2})^{2}$ ∎

######  Corollary A.1.

[Similar Manifolds] Consider random hyperplanes and spheres in $d$ dimensions
where all coefficient’s of hyperplanes are chosen as normal random variables
and for spheres the coordinates and radius chosen as unit normal random
variables. By looking at the similarity $T(U_{1}).T(U_{2})$ we get

1\. The expected similarity between two random lines is $1/d$

2\. Between two parallel lines is $1-O(1/d)$

3\. Between two random spheres is $1/5$

4\. Between two concentric spheres is $1$

5\. Between a random line and a random sphere is $O(1/d)$

###  A.2 Proof of Proposition 3.2

###### Proof.

For the signature of $U_{1}\cap U_{2}$, first consider the case when they
linear manifolds that are subpaces. Then $F(U)=I-T(U)$ is the projection
matrix for the subspace. $T(U_{1}\cap U_{2})$ will be the union of the null
spaces $T(U_{1})$,$T(U_{2})$ – this is because the intersection of the
concepts will satisfy any polynomial equation satisfied by either of the two
concepts. To obtain the projection matrix $F(U_{1}\cap U_{2})$ we note that
repeated multiplication of a vector by $F(U_{1})$ and $F(U_{2})$ will nullify
any component vector outside intersection of $U_{1}$ and $U_{2}$; if a vector
is in the intersection then the multiplications keeps it unchanged; otherwise
its length keeps decreasing till it ends up getting projected into $U_{1}\cap
U_{2}$. The same argument when applied on subspaces formed over $\phi(x)$
which extends this to general manifolds.

Next, let $v_{1},..,v_{k_{1}}$ be a basis for $U_{1}$ subspace and
$w_{1},..,w_{k_{2}}$ be a basis for $U_{2}$ subspace. Then $F(U_{1})=\sum
v_{i}v_{i}^{\top}$ and $F(U_{2})=\sum w_{i}w_{i}^{\top}$. Thus,
$F(U_{1}).F(U_{2})=\sum_{i,j}(v_{i}.w_{j})^{2}$. If $U_{1}$ and $U_{2}$
intersect in $\text{dim}(U_{1}\cap U_{2})$, then we can find basis that share
$\text{dim}(U_{1}\cap U_{2})$ basis vectors. This gives

$F(U_{1}).F(U_{2})\geq dim(U_{1}\cap U_{2}).$

For random $v_{i}$ and $w_{j}$, $E[v_{i}.w_{j}]=\frac{1}{d}$ which proves the
result for random subspaces $U_{1}$ and $U_{2}$ (i.e.,
$E[F(U_{1}).F(U_{2})]=k^{2}/d$).

The result for $T(U)$ follow from the fact that $T(U)=I-F(U)$.

∎

##  Appendix B Signature of translated and rotated image manifold

In this section, we show how rotating an image produces the signatures on an
analytic manifold which means we can use Theorem 5.2 to obtain a signature of
this manifold. Here, we explicitly show the Taylor series that transforms
signatures under rotation $\theta$.

######  Lemma B.1.

Let $X$ denote the point cloud of the pixels of image (each pixel can be
viewed as concatenation of position and color coordinates) and let
$X_{\theta}$ denote the rotation of the image under rotation $\theta$. Then,
$S(X_{\theta})$ can be written as a Taylor series in $\theta$. (We derive this
Taylor series upto the second order approximation).

###### Proof.

Note that each point goes through following transformation:

$\displaystyle x^{\prime}$ $\displaystyle=x\cdot
cos\theta+y\cdot\sin\theta\quad\text{and}\quad
y^{\prime}=-x\cdot\sin\theta+y\cdot\cos\theta,$

where point $x^{\prime}$ is obtained by applying the rotation matrix
corresponding to $\theta$ angle on point $x$. The second degree moment can be
computed as

$\displaystyle M_{x^{\prime}}$
$\displaystyle=\mathbb{E}[x^{\prime}]=M_{x^{\prime}}\cos\theta+M_{y^{\prime}}\sin\theta$
$\displaystyle M_{y^{\prime}}$
$\displaystyle=\mathbb{E}[y^{\prime}]=-M_{x}\sin\theta+M_{y}\cos\theta$
$\displaystyle M_{x^{\prime 2}}$ $\displaystyle=\mathbb{E}[x^{\prime
2}]=M_{x^{2}}\cos^{2}\theta+M_{y^{2}}\sin^{2}\theta+2M_{x}M_{y}\cos\theta\sin\theta.$
$\displaystyle M_{y^{\prime 2}}$ $\displaystyle=\mathbb{E}[y^{\prime
2}]=M_{x^{2}}\sin^{2}\theta+M_{y^{2}}\cos^{2}\theta-2M_{x}M_{y}\cos\theta\sin\theta$
$\displaystyle M_{x^{\prime}y^{\prime}}$
$\displaystyle=\mathbb{E}[x^{\prime}y^{\prime}]=-M_{x^{2}}\sin\theta\cos\theta+M_{y^{2}}\sin\theta\cos\theta+M_{x}M_{y}(\cos^{2}\theta-\sin^{2}\theta)$

For small rotation $\theta$, we can approximate $\sin\theta\approx\theta$ and
$\cos\theta\approx 1-\frac{\theta^{2}}{2}$ using taylor series with
approximation error of $O(\theta^{3})$. Using this approximation, we obtain:

$\displaystyle M_{x^{\prime}}$
$\displaystyle=M_{x}\Big{(}1-\frac{\theta^{2}}{2}\Big{)}+M_{y}\theta$
$\displaystyle M_{y^{\prime}}$
$\displaystyle=-M_{x}\theta+M_{y}\Big{(}1-\frac{\theta^{2}}{2}\Big{)}$
$\displaystyle M_{x^{\prime 2}}$
$\displaystyle=M_{x^{2}}(1-\theta^{2})+M_{y^{2}}\theta^{2}+2M_{xy}\theta$
$\displaystyle M_{y^{\prime 2}}$
$\displaystyle=M_{x^{2}}\theta^{2}+M_{y^{2}}(1-\theta^{2})-2M_{xy}\theta$
$\displaystyle M_{x^{\prime}y^{\prime}}$
$\displaystyle=-M_{x^{2}}\theta+M_{y^{2}}\theta+M_{xy}(1-2\theta^{2}).$

This second order moment allows us to use the power transformation to get the
signature of the rotated data manifold. Simplifying above equations, we obtain

$\displaystyle\left(\begin{array}[]{c}M_{x^{\prime}}\\\ M_{y^{\prime}}\\\
M_{x^{\prime 2}}\\\ M_{y^{\prime 2}}\\\
M_{x^{\prime}y^{\prime}}\end{array}\right)\approx\left(\begin{array}[]{c}M_{x}\\\
M_{y}\\\ M_{x^{2}}\\\ M_{y^{2}}\\\
M_{xy}\end{array}\right)+\theta\left(\begin{array}[]{c}M_{y}\\\ -M_{x}\\\
M_{xy}\\\ -M_{xy}\\\
M_{y^{2}}-M_{x^{2}}\end{array}\right)+\frac{\theta^{2}}{2}\left(\begin{array}[]{c}-M_{x}\\\
M_{y}\\\ 2M_{y^{2}}-2M_{x^{2}}\\\ 2M_{x^{2}}-2M_{y^{2}}\\\
-4M_{xy}\end{array}\right)$

∎

Next, we describe the manifold obtained by translation of a given image.

######  Lemma B.2 (Signature of translated image manifold).

Let $S(X_{(u,v)})$ be the signature of an image shifted by $(u,v)$. Then, all
such signatures $S(X_{(u,v)})$ over different shifts lies on an analytic
manifold.

###### Proof.

Suppose each point goes through the following transformation:

$\displaystyle x^{\prime}=x+u\quad\text{and}\quad y^{\prime}=y+v$

The moments after the translation is given by

$\displaystyle M_{x^{\prime}}$ $\displaystyle=M_{x}+u$ $\displaystyle
M_{y^{\prime}}$ $\displaystyle=M_{y}+v$ $\displaystyle M_{x^{\prime 2}}$
$\displaystyle=M_{x^{2}}+2M_{x}u+u^{2}$ $\displaystyle M_{y^{\prime 2}}$
$\displaystyle=M_{y^{2}}+2M_{y}v+v^{2}$ $\displaystyle
M_{x^{\prime}y^{\prime}}$ $\displaystyle=M_{xy}+M_{x}v+M_{y}u+u\cdot v$

Suppose the translation $u$ and $v$ are small such that the second order terms
are neglible (e.g., $u^{2},v^{2}$ and $u\cdot v$ are negligible), then we can
obtain the second order moment statistics as a linear approximation with
quadratic approximation error as follows:

$\displaystyle\left(\begin{array}[]{c}M_{x^{\prime}}\\\ M_{y^{\prime}}\\\
M_{x^{\prime 2}}\\\ M_{y^{\prime 2}}\\\
M_{x^{\prime}y^{\prime}}\end{array}\right)\approx\left(\begin{array}[]{c}M_{x}\\\
M_{y}\\\ M_{x^{2}}\\\ M_{y^{2}}\\\
M_{xy}\end{array}\right)+u\left(\begin{array}[]{c}1\\\ 0\\\ 2\cdot M_{x}\\\
0\\\ M_{y}\end{array}\right)+v\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\ 2\cdot
M_{y}\\\ M_{x}\end{array}\right)$

∎

##  Appendix C Proofs from Section 5

###  C.1 Explicit polynomial representation from implicit polynomials

We will first prove the case when $X$ is a $k-$dimensional manifold in
$(k+1)-$dimensional space.

######  Lemma C.1.

Suppose $X\in\mathbb{R}^{k+1}$ can be represented as a degree $l$ polynomial
in terms of implicit random variables $z\in\mathbb{R}^{k}$. i.e., $X=G(z)$
where the degree of $p$ is $r$, then there exists a polynomial in $X$, $H(X)$,
of the degree $(cr)^{k}$ for some absolute constant $c$ such that $H(X)=0$ for
points on the manifold.

###### Proof of Lemma C.1.

Let $u$ be the upper bound on the degree of $H(X)$. We show that there exists
a polynomial constructed from the entries of $X^{\otimes u}$ such that
$H(X)=0$ for points on the manifold.

We have $X^{\otimes u}=G(z)^{\otimes u}$ by taking $u^{\text{th}}$ tensor
power of $X=G(z)$ equality. Denote the number of different monomial entries in
$X^{\otimes u}$ with $X\in\mathbb{R}^{k+1}$ by $m(u,k+1)$. Observe that
$X^{\otimes u}=G(z)^{\otimes u}$ has $m(u,k+1)$ different equalities from
because $X^{\otimes u}$ has $m(u,k+1)$ different monomials. The right side
$G(z)^{\otimes u}$ has at most $m(u\cdot r,k)$ different monomial terms, and
when $m(u,k+1)>m(u\cdot r,k)$, we can eliminate the implicit variables in the
remaining $m(u,k+1)-m(u\cdot r,k)$ equations. The remaining equations are
consistent because $X$ is generated using $G(z)$.

The degree of the polynomial $\phi(X)$ is the smallest $u$ such that
$m(u,k+1)>m(u\cdot r,k)$. The number of monomials of degree $u$ in $k+1$
variables is given by $m(u,k+1)=C^{u+k}_{u}$. The degree of $\phi(X)$ is
smallest $u$ such that $C^{u+k}_{k}>C^{r\cdot u+k-1}_{k-1}$. This inequality
satisfies when $u>(cr)^{k-1}$ for some absolute constant $c$.

$\displaystyle C^{r\cdot u+k-1}_{k-1}$ $\displaystyle\leq\frac{e^{k-1}(r\cdot
u+k-1)^{k-1}}{(k-1)^{k-1}}\leq\frac{(2e)^{k-1}(r\cdot
u)^{k-1}}{(k-1)^{k-1}}\leq\Big{(}\frac{2e}{c(k-1)}\Big{)}^{k-1}u^{k}$
$\displaystyle\leq\Big{(}\frac{1}{k^{k}}\Big{)}(u+k)^{k}\leq C^{u+k}_{k}$

The second inequality follows from $k-1\leq r\cdot u$. ∎

###### Proof of Theorem 5.1.

For extending the Lemma C.1 to $X\in\mathbb{R}^{d}$ case, we start by
identifying $k$ direction along a local neighbourhood of any point on the
neighbourhood. Note that this can be done by looking at the tangent direction
of the manifold of the points in the neighbourhood. After determining the
tangent direction, we can rotate the points so that
$\\{X_{1},X_{2},\ldots,X_{k}\\}$ in the neighbourhood.

Assuming $\\{X_{1},X_{2},\ldots,X_{k}\\}$ covers a $k-$dimensional space, we
use Lemma C.1 for each of $\\{X_{k+1},\ldots,X_{d}\\}$ as $k+1$-th coordinate
and obtain $d-k$ polynomials of the degree $(cl)^{k}$ for some absolute
constant $c$. Note that each of the polynomials contains a feature mapping
$H(X)$ that depends on the first $k$ coordinates of $X$ and one of the last
$d-k$ coordinates such that $H(X)=0$ for points on the manifold. ∎

###  C.2 Approximate polynomial representation for analytic manifolds

We will first prove for the case when $X$ is a $k-$dimensional manifold in a
$(k+1)-$dimensional space given by the analytic function.

######  Lemma C.2.

Suppose the data $X\in[-1,1]^{k+1}$ lies in a $k-$dimensional manifold by
equation $X=G(z)$ where $G:[-1,1]^{k}\to[1,1]^{k+1}$ is an analytic function
with bounded gradient $\|\nabla^{(m)}G(x)\|\leq 1$. Then, there exists a
polynomial in $X$, $H(X)$ of degree $u$ such that for points on the manifold,
$|H(X)|\leq u^{-\frac{u^{(1+1/k)}}{2c^{k}k}}$.

###### Proof of Lemma C.2.

Since $G(z)$ is an analytic function with the bounded derivatives so we can
approximate $G(z)$ using a degree $q$ Taylor approximation $G^{\prime}(z)$
within the ball of a unit radius of $z$ with the approximation error
$\frac{1}{q!}$. Now, instead of Taylor expanding each term in $G(z)$, we can
expand each term in $G(z)^{\otimes l}$. Suppose for any multi-index
$\alpha=(\alpha_{1},\ldots,\alpha_{l})$, the Taylor approximation of
$G_{\alpha}(z)=\prod_{i=1}^{l}G_{\alpha_{i}}(z)$ of degree $q$ is given by
$\tilde{G}_{\alpha}(z)$ as follows:

$\displaystyle\tilde{G}_{\alpha}(z)=G_{\alpha}(\mathbf{0})+\sum_{i=1}^{q-1}\frac{\langle\nabla^{(i)}G_{\alpha}(\mathbf{0}),z^{.i}\rangle}{i!}.$

where $\mathbf{0}$ represents the zero vector. Using the remainder theorem of
the Taylor series, we have

$\displaystyle|\tilde{G}_{\alpha}(z)-G_{\alpha}(z)|\leq\frac{\|\nabla^{(i)}G_{\alpha}(\mathbf{0})\|}{q!}\leq\frac{l^{q}}{q!}.$

where the last inequality follows from the fact that the $i^{\text{th}}$
derivative of $\tilde{G}_{\alpha}(z)$ for any $\alpha$ can be written as a sum
of $l^{i}$ terms with each term bounded by 1. Hence, we can approximate each
term in $G(z)^{\otimes l}$ with $\tilde{G}$ within the ball of the unit radius
with the approximation error $l^{q}/q!$ where each element of
$\tilde{G}(z)^{(.u)}$ is a degree $q-$polynomial obtained by Taylor expansion.
Recall that the $m(l,k)$ is defined as the number of different monomials of
degree $l$ in $k$ variables. The total number of variables in $\tilde{G}(z)$
is given by $m(q,k)$ and the total number of different equations in
$X^{.u}=G(z)^{.u}$ is $m(u,k+1)$. The degree of polynomial $H(X)$ is the
smallest $l$ such that $m(u,k+1)>m(q,k)$. This inequality reduces to
$C^{u+k}_{k}>C^{q+k-1}_{k-1}$. Choosing $u=c^{k-1}q^{\frac{k-1}{k}}$ for some
absolute constant $c$, then this inequality holds.

$\displaystyle
C^{q+k-1}_{k-1}\leq\Big{(}\frac{e(q+k-1)}{(k-1)}\Big{)}^{k-1}\leq\frac{u^{k}}{k^{k}}\leq\frac{(u+k)^{k}}{k^{k}}\leq
C^{u+k}_{k}$

Next, we show that the polynomial is close to zero for the points on the
manifold. The polynomial obtained by a Taylor expansion of the analytic
function $G$ with the truncation at $q^{th}$ degree has approximation
$\frac{u^{q}}{q!}$. Putting the value of $q=\frac{u^{k/(k-1)}}{c^{k}}$, we
obtain the approximation error to be
$\frac{u^{q}}{q!}\leq\frac{c^{q}u^{q}}{q^{q}}=c^{q}u^{-\frac{q}{k}}=u^{-\frac{u^{(1+1/k)}}{2c^{k}k}}$.
∎

###### Proof of Theorem 5.2.

Similar to proof of Theorem 5.1, we start by rotating the coordinates at each
point such that the first $k$ coordinates $\\{X_{1},X_{2},\ldots,X_{k}\\}$ in
the neighbourhood covers $k$ dimensional space. Then, we use Lemma C.2 for
each of $\\{X_{k+1},\ldots,X_{d}\\}$ as $k+1$-th coordinate and obtain $d-k$
polynomials that satisfies the given property. ∎

###  C.3 Quantitative bounds for checking membership

We will use the following classical result about multi-variate polynomials:

######  Lemma C.3 (Anti-concentration for polynomials, [CW01]).

Let $P:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a degree at most $d$
polynomial. Let $x$ be a random point in the unit ball. Let
$\|P\|_{2}=E_{x}[P(x)^{2}]^{1/2}$. Then, for any $t,\epsilon$,

$Pr_{x}[|P(x)-t|<\epsilon\|P\|_{2}]=O(\epsilon^{1/d}).$

The following lemma says if we start from a polynomial with non-trivial norm,
then for any ball that is centered not too far from the origin, the polynomial
is unlikely to be very small. It will be important that this probability goes
to zero as the radius of the ball increases.

######  Lemma C.4.

Let $Q:\mathbb{R}^{k}\rightarrow\mathbb{R}$ be a degree $s$ polynomial such
that the sum of squares of the coefficients is $1$. Then, for any
$y\in\mathbb{R}^{k}$, if $\|y\|\leq R$, and $z$ a uniformly random point in
the unit ball of $\mathbb{R}^{k}$,

$Pr[|Q(y+\Delta z)|>\epsilon]<(ks)\epsilon^{1/s}\alpha^{1/s},$

where
$\alpha=\max((1+\|y\|_{\infty})/\Delta,((1+\|y\|_{\infty})/\Delta)^{s})$.

###### Proof Sketch for Lemma C.4.

Let $P(z)=Q(y+\Delta z)$. We first argue that $E_{z}[P(z)^{2}]$ is non-
trivially large. First, observe that $E[Q(z)^{2}]=s^{-O(s)}$ as $Q$’s sum-of-
squares of coefficients is $1$.

Without loss of generality, let us suppose $P$ has no constant term. Now,
suppose that $E[P(z)]^{2}=\delta$. Let $P(z)=\sum_{S}c_{S}h_{S}(z)$ where
$h_{S}$ are the Legendre polynomials that form an orthonormal basis for
polynomials under the uniform distribution on the sphere. Thus,
$\sum_{S}c_{S}^{2}=\delta^{2}$.

Now, for $z^{\prime}$ uniformly random on the unit ball, let
$Q(z^{\prime})=P((z^{\prime}-y)/\Delta)$. Therefore,
$E[Q(z^{\prime})^{2}]=E[(\sum_{S}c_{S}h_{S}((z^{\prime}-y)/\Delta))^{2}]$.

However, note that the Legendre polynomials of degree $s$ are bounded: in
particular for any point $x$, and $h_{S}$ of degree $s$, we have
$|h_{S}(x)|\leq s^{O(s)}(\|x\|_{\infty}+\|x\|_{\infty}^{s})$.

Therefore, if we let
$\alpha=\max((1+\|y\|_{\infty})/\Delta,((1+\|y\|_{\infty})/\Delta)^{s})$, we
get

$E[Q(z^{\prime})^{2}]\leq(\sum_{S}|c_{S}|)\cdot s^{O(s)}\alpha\leq
k^{O(s)}s^{O(s)}\alpha\delta.$

Therefore, we get $\delta>(ks)^{-O(s)}/\alpha$.

Finally, applying the previous lemma, we get

$Pr_{z}[|P(z)|<\epsilon]=O((\epsilon/\delta)^{1/s}).$

Therefore, $Pr_{z}[|Q(y+\Delta
z)|<\epsilon]<(ks)\cdot\alpha^{1/s}\epsilon^{1/s}$.

Finally, note that
$\alpha=\max((1+\|y\|_{\infty})/\Delta,((1+\|y\|_{\infty})/\Delta)^{s})$ is a
decreasing function in $\Delta$. ∎

##  Appendix D Proofs for Section Section 6

###### Proof.

Without loss of generality, we assume that $E[X_{i}^{2}]=1$. By definition,
$\hat{G}_{1,j}=\frac{1}{N}\sum_{i=1}^{N}\sigma(x_{i}\cdot
r_{j})=\frac{1}{N}\sum_{i=1}^{N}(r_{j}^{\top}x_{i}x_{i}^{\top}r_{j})=r_{j}^{\top}M(X)r_{j}$.
The expectation of $\hat{G}_{1,j}$ with respect to random Gaussian weights
$r_{j}$ can be simplified using Stein’s lemma. Therefore, we have

$\displaystyle\mathbb{E}_{r}[r^{\otimes
2}(r^{\top}M(X)r)]=\mathbb{E}_{r}[\nabla^{(2)}(r^{\top}M(X)r)+(r^{\top}M(X)r)I]=M(X)+(d+1)I.$

Similarly, simplifying $r_{j}^{\otimes 2}\hat{G}_{2,j}$ with respect to random
Gaussian weights $r_{j}\sim\mathcal{N}(0,I)$, we have

$\displaystyle\mathbb{E}[r^{\otimes 2}\hat{G}_{2,j}]$
$\displaystyle=\mathbb{E}[\nabla^{(2)}(r^{\top}M(X)r)^{2}+(r^{\top}M(X)r)^{2}]$
$\displaystyle=\mathbb{E}[2(r^{\top}M(X)r)(M(x)+I)+(M(X)+I)rr^{\top}(M(X)+I)+(r^{\top}M(X)r)^{2}I].$
$\displaystyle=2d(M(X)+I)+M(X)^{2}+2M(X)+I+3\sum_{i=1}^{d}M_{i,i}(X)^{2}I+\sum_{i\neq
j}M_{i,i}(X)M_{j,j}(X)I$ $\displaystyle=M(X)^{2}+(2d+2)M(X)+(d^{2}+4d+1)I$

Setting $\beta_{1}=-(2d+2)$ and $\beta_{2}=d^{2}+1$, we obtain the result. ∎


# Fabry-Perot Interferometric Calibration of 2D Nanomechanical Plate
Resonators

Myrron Albert Callera Aguila<EMAIL_ADDRESS>Department of
Engineering and System Science, National Tsing Hua University, Hsinchu, 30013,
Taiwan  Nano Science and Technology Program, Taiwan International Graduate
Program, National Tsing Hua University and Academia Sinica  Institute of
Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan  Joshoua Condicion
Esmenda<EMAIL_ADDRESS>Department of Engineering and System
Science, National Tsing Hua University, Hsinchu, 30013, Taiwan  Nano Science
and Technology Program, Taiwan International Graduate Program, National Tsing
Hua University and Academia Sinica  Institute of Physics, Academia Sinica,
Nankang, Taipei, 11529, Taiwan  Jyh-Yang Wang  Institute of Physics, Academia
Sinica, Nankang, Taipei, 11529, Taiwan  Teik-Hui Lee  Institute of Physics,
Academia Sinica, Nankang, Taipei, 11529, Taiwan  Chi-Yuan Yang  Institute of
Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan  Kung-Hsuan Lin
Institute of Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan  Kuei-
Shu Chang-Liao  Department of Engineering and System Science, National Tsing
Hua University, Hsinchu, 30013, Taiwan  Sergey Kafanov  Department of Physics,
Lancaster University, Lancaster, LA1 4YB, United Kingdom  Yuri A. Pashkin
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United
Kingdom  Chii-Dong Chen<EMAIL_ADDRESS>Institute of Physics,
Academia Sinica, Nankang, Taipei, 11529, Taiwan

###### Abstract

Displacement calibration of nanomechanical plate resonators presents a
challenging task. Large nanomechanical resonator thickness reduces the
amplitude of the resonator motion due to its increased spring constant and
mass, and its unique reflectance. Here, we show that the plate thickness,
resonator gap height, and motional amplitude of circular and elliptical drum
resonators, can be determined in-situ by exploiting the fundamental
interference phenomenon in Fabry-Perot cavities. The proposed calibration
scheme uses optical contrasts to uncover thickness and spacer height profiles,
and reuse the results to convert the photodetector signal to the displacement
of drumheads that are electromotively driven in their linear regime.
Calibrated frequency response and spatial mode maps enable extraction of the
modal radius, effective mass, effective driving force and Young’s elastic
modulus of the drumhead material. This scheme is applicable to any
configuration of Fabry-Perot cavities, including plate and membrane
resonators.

Nanomechanical resonators (NMRs) are exceptional force and mass sensors[1, 2],
which made them valuable test platforms for the investigation of various
phenomena at the nanoscale such as synchronization[3, 4], noise[5, 6],
nonlinearity[7, 8], and light-matter interaction[9, 10, 11]. NMRs with
flexural modes (i.e. plates and beams) have attracted interest due to their
linear response even to large deformation-inducing forces[12, 13]. The well-
known mechanical properties of the bulk material and its geometry determine
plate and beam frequencies that can be predicted to a high accuracy. This
allows the realization of unique device applications such as nanomechanical
mass spectrometers[14, 15], phononic crystals built from NMR arrays[16, 17],
and complex networks of NMRs embedded in electrical circuits[3, 4], and
cavity-mediated quantum systems[18].

Calibration of NMR displacement is important for quantification of device
characteristics in sensing applications. While the spatial dynamics of
membrane NMRs, whose behavior is to a large extent determined by tensile
stress, have been investigated in great detail with optical interferometry[19,
20, 21, 8], plate NMRs have been less explored. Studies involving Fabry-Perot
(FP) cavities have introduced calibration of the vibrational amplitudes of
membrane and string NMRs[22, 20, 23, 24]. However, they are hardly applicable
to plate NMRs because of reduced vibrational amplitudes owing to the increased
spring constant and mass, and unique reflectance versus FP cavity length
arising from thick absorptive materials such as niobium diselenide. Also,
there are cases where a smaller spacing is preferred over the optimal spacing
for interferometric detection. These cases include mechanical frequency tuning
by low gate voltages[25, 26, 27], and stronger electomechanical coupling
between mechanical resonators and microwave cavities[28, 29].

In this Letter, we show that the motion of plate NMRs can be calibrated by
considering multilayer wave interferences occurring on the FP structure. To
demonstrate the robustness of the technique, a thick 2D material, NbSe2, is
used as the drumhead. This approach allows determination of the layer
thickness, spacer height and device responsivity of each translucent flexible
mirror. Our calibration scheme reveals subnanometer mechanical displacements
for the measured response of plate NMRs with thickness exceeding 50 nm.

Figure 1(a) shows Device A, a circular plate with a hole diameter of
7$\,\mu$m, and Device B, an elliptical plate with hole diameters of 8$\,\mu$m
(X, major axis) and 7$\,\mu$m (Y, minor axis). The devices share the same
flake, ground electrode, and driving voltages. The NbSe2 flake and ground
electrodes are separated by the insulating layer (electron-beam resist
CSAR-62) and vacuum spacers, and hence form FP cavities for detection, and
capacitors for actuation. A large rectangular opening, located tens of microns
below the drum centers, allows the flake to connect to the voltage-controlled
Au/Cr electrodes. The motion of the electromotively driven plates is detected
interferometrically in a high vacuum environment[30].

Figure 1: (a) Optical micrograph of the NbSe2 plate resonator devices. The
circuit diagram shows the electromotive scheme used in driving the resonator.
(b) Schematic drawing of the optical cross-section of the device (zones 3 and
4) and the references (zones 1 and 2) as measured via Fabry-Perot
interference. The actuation circuit is added for clarity. (c) Confocal image
showing devices A and B as scanned with a probe laser beam with wavelength
$\lambda=532\,$nm. Scale bars in (a) and (c) are set to $10\,\mu$m. (d) Sketch
of the output voltage of the fast photodetector versus time of a driven
flexural resonator.

Our method relies on different contrast of light elastically reflected from
each zone, as shown in Fig. 1(b). The flake (pink bar) acts as a translucent
movable mirror with thickness $h$, which is separated from the ground
electrode by a spacer of height $s$. For convenience, the reflected intensity
is expressed in terms of the reflectance $R$, which is the ratio of the total
reflected light intensity to the incident intensity. Stationary mirrors have
only DC component $R=\overline{R}$ while movable mirrors have both
$\overline{R}$ and AC component $\widetilde{R}$. Zones 1 and 2 represent two
stationary mirrors: stacks of gold, orange, green and blue bars having
reflectance $R_{1}=\overline{R}_{1}$ and a mirror covered with a spacer (light
gray) having reflectance $R_{2}=\overline{R}_{2}$, respectively. Zone 3
represents two stationary mirrors separated by a dielectric gap (clamp) with
reflectance $R_{3}=\overline{R}_{3}$. Finally, zone 4 is the main FP cavity
composed of one stationary and one movable mirrors, which are separated by a
vacuum gap with reflectance $R_{4}$. Here, zones 1 and 2 are references for
zones 4 and 3, respectively. Scanning mirrors in the measurement setup move
the laser spot in each zone a distance $X$ and $Y$ away from the drums’
center.

Application of DC and AC voltages to the flake exerts an attractive force; the
NMR responds with an out-of-plane motional amplitude $z$ at a driving
frequency $f_{d}$. Due to the position and motion of the movable mirror in
zone 4, the main FP cavity has reflectance
$R_{4}=\overline{R}_{4}+\widetilde{R}_{4}(f_{d})$, with
$\overline{R}_{4}\gg\widetilde{R}_{4}(f_{d})$. Figure 1(d) shows the
photodetector output signal $V$ acquired from $R_{4}$. Both the DC component
$\overline{V}$ and the AC component $\widetilde{V}$ of the output signal are
proportional to $\overline{R}_{4}$ and $\widetilde{R}_{4}$, respectively.
Amplitude $z$ is determined after obtaining $h$ and $s$.

Though we calculate $\overline{R}_{1-4}$ using the multilayer interference
approach[31, 32, 33] (MIA), the reflectance of FP cavities with four
interfaces[34, 35] $\overline{R}_{3,4}$ captures the stationary reflections
occurring for each drum. Here, we assume that the coherent probe light, having
wavelength $\lambda$, originates from a point source and propagates from a
semi-infinite vacuum layer. The drum and the bottom mirror have complex
refractive indices $\hat{n}_{h}$[36] and $\hat{n}_{m}$, respectively, whereas
the spacers have real refractive index $\hat{n}_{s}$ ($\hat{n}_{s,drum}$ for
the vacuum spacer and $\hat{n}_{s,clamp}$ for the CSAR-62 spacer). In this
geometry, the vacuum-NMR, NMR-spacer, and spacer-mirror interfaces contribute
significantly to the cavity’s overall reflectance. The total reflectance is
defined as

${\overline{R}_{3,4}}={\left|{\frac{{{r_{h}}+{r_{s}}{e^{-2j{\delta_{h}}}}+\left[{{r_{h}}{r_{s}}+{e^{-2j{\delta_{h}}}}}\right]{\Gamma_{m}}{e^{-2j{\delta_{s}}}}}}{{1+{r_{h}}{r_{s}}{e^{-2j{\delta_{h}}}}+\left[{{r_{s}}+{r_{h}}{e^{-2j{\delta_{h}}}}}\right]{\Gamma_{m}}{e^{-2j{\delta_{s}}}}}}}\right|^{2}}$
(1)

where $\delta_{h}=2\pi\hat{n}_{h}h/\lambda$ is the optical phase thickness of
the NMR, $\delta_{s}=2\pi\hat{n}_{s}s/\lambda$ is the optical phase thickness
of the spacer, $r_{h}$=$(1-\hat{n}_{h})/(1+\hat{n}_{h})$ is the Fresnel
coefficient of the vacuum-NMR interface,
$r_{s}$=$(\hat{n}_{h}-\hat{n}_{s})/(\hat{n}_{h}+\hat{n}_{s})$ is the Fresnel
coefficient of the NMR-spacer interface, and
$\Gamma_{m}=(\hat{n}_{s}-\hat{n}_{m})/(\hat{n}_{s}+\hat{n}_{m})$ is the
equivalent Fresnel coefficient of the spacer-mirror interface. For
convenience, $\Gamma_{m}$ is computed using MIA[30].

Figure 1(c) shows the topographical features of the drum devices as probed by
a continuous wave laser beam. Apparently, the reflectance signal measured
along the white dashed lines drawn across devices A and B contains
$\overline{V}_{3}$, taken outside the dashed ellipses, and $\overline{V}_{4}$,
taken within the dashed ellipses. Polygons 1 and 2 give average values of
$\overline{V}_{1}$ and $\overline{V}_{2}$. The colored dashed ellipses,
representing the hole diameters measured in Fig. 1(a), are smaller than the
light gray ellipses. These gray ellipses manifest in Fig. 1(a) as concentric
purple rings seen for each drum. These features arise when the flake transfer,
that is based on elastomeric stamps, deforms the edge of every drum hole.

Figure 2: (a) Diagram for determining $h$ and $s$ for the clamp and drum
zones. Minimization of the difference between the experimental contrast
($C_{\mathrm{exp}}$) and the modelled contrast ($C_{\mathrm{mod}}$) results in
$h$ and $s$ profiles for Device A (b) and Device B (c). Colored dashed lines
refer to the hole radius set in Figs. 1(a,c), separating the drum (white
fills) and clamp (gray fills) zones.

Since $\overline{R}$ is susceptible to scattering losses[37], we circumvent
this issue by normalizing the Michelson contrast[34] of each FP cavity to
their reference. Having defined the experimental and calculated reflectance,
the cavity’s optical contrast, $C$, is quantified as
$C=(\overline{R}_{\mathrm{3,4}}-\overline{R}_{\mathrm{2,1}})/(\overline{R}_{\mathrm{3,4}}+\overline{R}_{\mathrm{2,1}})$,
where $\overline{R}_{\mathrm{3,4}}$ is the stationary reflectance of the FP
cavity, and $\overline{R}_{\mathrm{2,1}}$ is the stationary reflectance of the
cavity’s reference. Apparently, $C$ ranges between -1 and 1, with zero
denoting no difference with the reference. If $C$ is positive, then the cavity
is brighter than the reference. Otherwise, the cavity is darker than its
reference.

The output voltages measured for each pixel along the dashed lines in Fig.
1(c) are converted into contrast values for devices A and B, as depicted in
Fig. 2(a). The experimental contrast $C_{\mathrm{exp}}$ represents the ratio
of voltages acquired from different zones in the confocal image of each device
while the modelled contrast $C_{\mathrm{mod}}$ is derived using MIA[30].
Figures 2(b-c) show the resulting $h$ and $s$ cross-section profiles acquired
from minimizing the difference between the experimental contrast values and
the contrasts generated by MIA. The mean plate thicknesses and spacer heights
for the two devices are in excellent agreement with the mean values listed in
Table 1. The spacer height for both drums and clamps agrees well with the
stylus profilometer measurements. From the flake thickness of about 55 nm, we
deduce 92 layers of NbSe2 sheets assuming a single layer thickness of
$0.6\,$nm[38].

Table 1: Mean flake and spacer thicknesses of NbSe2 drum and clamp zones  Devices | A | B
---|---|---
$h_{drum}$ (nm) | 55.139 $\pm$ 0.002 | 55.135 $\pm$ 0.002
$s_{clamp}$ (nm) | 55.03 $\pm$ 0.05 | 55.05 $\pm$ 0.04
$s_{drum}$ (nm) | 297.2 $\pm$ 0.1 | 297.3 $\pm$ 0.1
$s_{drum}$ (nm) | 296.0 $\pm$ 0.3 | 295.9 $\pm$ 0.3

The $h$ profiles in Figs. 2(b-c) show a hundred picometer variation between
the drum and clamp zones. Meanwhile, buckling is observed in the $s$ profiles
in Figs. 2(b-c) as $s_{drum}$ for both devices are greater than $s_{clamp}$ by
$1.2-1.4\,$nm. We see the drumheads bulge[39, 40] presumably due to the
pressure difference between the trapped air in the drum hole and the vacuum
environment. The surface roughness of the movable mirror likely comes from the
thermally-grown oxide[41] on the surface of the stationary mirror.

Having determined the mean $h_{drum}$ and $s_{drum}$, we evaluate the optical
reflectance-to-displacement responsivity $\left|d\overline{R}_{4}/ds\right|$
of each drum. This quantity is obtained from $\widetilde{R}_{4}(f_{d})$ =
$\left|d\overline{R}_{4}/ds\right|z(f_{d})$. The AC component reflected from
zone 4 and characterized by $\widetilde{R}_{4}$, being purely due to
mechanical motion, is insensitive to any scattering losses as this wave is
amplitude-modulated. Eq. (1) is then corrected by a prefactor that accounts
for the finite spot size of the probe Gaussian beam[30].

Figure 3: (a) Waterfall plot of FP reflectivity as a function of $\lambda$ at
varying $s$, with the used probe wavelength (green plane) situated at
$\lambda=532\,$nm. (b) Colored scatter plot of $\lambda_{FP}$ as a function of
$s$. The slope of the red solid line originates from the intersection of the
red plane with the $\lambda-s$ plane in (a). (c) Colored scatter plot of
$\overline{R}_{4}$ as a function of the cavity shift $\lambda_{FP}$. The blue
solid line comes from the the intersection of the blue plane with the red
plane in (a).

We define the average $\left|d\overline{R}_{4}/ds\right|$ to account for
spatial variations in $s_{drum}$ across the plate due to the pressure
difference and DC voltage. Note that each complex-valued refractive index is
dependent on the probing wavelength; this translates into the wavelength-
dependent $r_{h}$, $r_{s}$ and $\Gamma_{m}$. We modeled
$\left|d\overline{R}_{4}/s\right|_{avg}$ by the chain rule
$\left|{\Delta{{\bar{R}}_{4}}\left(\lambda\right)}/{\Delta{\lambda_{FP}}}\right|{\left|{\Delta{\lambda_{FP}}}/{\Delta
s}\right|_{s=s_{drum}}}$, where $\Delta\overline{R}_{4}/\Delta\lambda_{FP}$ is
the change of $\overline{R}_{4}$ with regards to the wavelength shift in the
FP cavity, and $\Delta\lambda_{FP}/\Delta s$ is the wavelength shift of the FP
cavity caused by the change of the spacer gap. The resulting dependences are
shown as a waterfall plot in Fig. 3(a) with a gap range exceeding the
uncertainty of our stylus profilometer[30]. Figure 3(a) demonstrates larger
$\overline{R}_{4}$ at near-infrared wavelengths. Figure 3(b) shows the peak
wavelength of the cavity, falling in the near-infrared range, shifting
linearly with a slope of $1.543\,$nm/nm as $s_{drum}$ increases from $285\,$nm
to $305\,$nm. Figure 3(c) shows how the shift consequently increases
$\overline{R}_{4}\left(\lambda\right)$ linearly, with a slope of $0.28\times
10^{-6}\,$nm-1. The product of these two slopes,
$\left|d\overline{R}_{4}/ds\right|_{avg}=0.43\times 10^{-3}\,$nm-1, agrees
with $\left|d\overline{R}_{4}/ds\right|_{s=s_{drum}}=0.40\times 10^{-3}\,$nm-1
that is evaluated from the gradient of the $\overline{R}_{4}$ with respect to
$s$[30]. The linear behavior seen in Figs. 3(b-c) is in contrast to the non-
linear dependence observed for optically-thin membranes in the same ranges of
$s$[30].

We use the average responsivity together with the interferometer system gain
$S$($\lambda$) (V/W), and the laser probe power $P_{in}$ to define the
displacement amplitude $z$ as

$z\left({{f_{d}},X,Y}\right)=\frac{{\widetilde{V}_{pk}\left({{f_{d}},X,Y}\right)}}{{\left|d\overline{R}_{4}\left({{\lambda},{h_{drum}},{s_{drum}}}\right)/ds\right|S\left({{\lambda}}\right){P_{in}}}}$
(2)

where $\widetilde{V}_{pk}$ is the frequency and position-dependent peak
voltage response of the NMR. The denominator in Eq. (2), when squared,
represents the transduction factor $\alpha$ (in V2/m2) that one can deduce
from the measured Brownian motion of a mechanical resonator probed by an
interferometric system[22, 20]. This quantity accounts for the device
responsivity and the detection parameters in the interferometric setup[30]. We
deduce transduction factors of $0.20\,\mu$V/pm for device A, and
$0.22\,\mu$V/pm for device B for probe powers listed in Figs. 4(a-b).

Figures 4(a-b) show the measured voltage response and its corresponding $z$
for devices A and B. The measured $z$ response profile agrees well with a
driven mechanical resonator model in the linear regime[42]:

$z\left({{f_{d}},X,Y}\right)=\frac{{{A_{eff}}}}{{4{\pi^{2}}\sqrt{{{\left({f_{d}^{2}-f_{m}^{2}}\right)}^{2}}+{{\left({{{{f_{d}}{f_{m}}}\mathord{\left/{\vphantom{{{f_{d}}{f_{m}}}{{Q_{m}}}}}\right.\kern-1.2pt}{{Q_{m}}}}}\right)}^{2}}}}}\Phi\left({X,Y}\right)$
(3)

where $f_{m}$ is the fundamental mode frequency, $Q_{m}$ is the mode quality
factor, and $A_{eff}$ is the amplitude expressed as effective acceleration.
$\Phi(X,Y)$ is the mode shape of the plate described as

$\Phi\left({X,Y}\right)={K_{0}}\left[{{J_{0}}\left({\beta
k(X,Y)}\right)-\frac{{{J_{0}}\left(\beta\right)}}{{{I_{0}}\left(\beta\right)}}{I_{0}}\left({\beta
k\left({X,Y}\right)}\right)}\right]$ (4)

where $J_{0}$ and $I_{0}$ are the zeroth Bessel and modified Bessel functions
of the first kind, respectively, $\beta=3.1961$ is the fundamental mode
constant for a clamped circular plate, and $K_{0}=0.947$ is a normalization
constant. Here,
$k\left({X,Y}\right)=\sqrt{{{\left({{X\mathord{\left/{\vphantom{Xa}}\right.\kern-1.2pt}a}}\right)}^{2}}+{{\left({Y/b}\right)}^{2}}}$
represents the normalized coordinates away from the maximum of $z$, where $a$
and $b$ represent the NMR major (X axis) and minor (Y axis) modal radii,
respectively. By setting $\Phi(X,Y)=1$, we measure $z_{A}=158\,\pm\,2\,$pm for
device A, and $z_{B}=259\,\pm\,3\,$pm for device B. Their magnitudes are three
orders of magnitude smaller than $h_{drum}$ and $s_{drum}$.

Figure 4: Spot-based displacement amplitude response (⚫) of device A (a) and device B (b) at $V_{DC}$ = 4 V and $V_{AC}$ = 125 mV and their driven resonator fits (red solid lines). The data shown in (c) and (d) refer to the spatial amplitude mode shape of devices A and B at $f_{d}=f_{m}$. The amplitude profile along X (⚫) and Y (⚫) axes of the mode shapes for devices A (e) and B (f) is fitted with a clamped circular plate model (red and blue solid lines for X and Y, respectively). Dashed vertical lines indicate the edges of the holes. Table 2: Modal properties of NbSe2 devices  Specifications | Device A | Device B | Method
---|---|---|---
$z_{max}$ (pm) | 161 | 320 | Eq. (4)
$a$ ($\mu$m) | 2.7 $\pm$ 0.2 | 3.19 $\pm$ 0.06 | Eq. (4)
$b$ ($\mu$m) | 2.6 $\pm$ 0.2 | 2.66 $\pm$ 0.02 | Eq. (4)
$m_{eff}$ (fg) | 1.4 | 1.74 | [30]
$A_{eff}$ (km/s2) | 132 | 132 | Eq. (3)
$F_{eff}$ (pN) | 191 | 229 | [30]
$E_{Y}$ (GPa) | 135 $\pm$ 13 | [30]

By driving the plates at $f_{m}$, and probing their spatial mode shape with
scanning mirrors, we observe surface plots of $z$ for devices A and B as shown
in Figs. 4(c-d). Figures 4(e-f) show X and Y axes cuts, with both axes
intersecting at $z_{max}$ of Figs. 4(c-d). They reveal $z$ profiles that agree
with Eq. (4), with $a$ and $b$ acting as free parameters. $z_{max}$, $a$, and
$b$ of the two plates are listed in Table 2. The discrepancy in the values of
$z_{B}$ and $z_{max}$ of device B is due to the location of the laser spot
that probed Fig. 4(b). Whereas $z_{A}$ lies at $X=Y\approx\,$0, $z_{B}$ lies
at $X,Y\approx 1\,\mu$m away from the spatial peak. Both $a$ and $b$ for
devices A and B are smaller than the hole radii (set as cyan and yellow dashed
lines in Figs. 4(e-f)), making $f_{m}$ for both devices higher than the
designed values. Moreover, device A shows unexpected elliptical modal behavior
with a miniscule difference between $a$ and $b$, which is due to the
fabrication process.

Table 2 lists other NMR-related quantities that are derived from Figure 4 such
as the effective mass $m_{eff}$, acceleration $A_{eff}$, force $F_{eff}$, and
Young’s elastic modulus $E_{Y}$. These quantities are derived from a clamped
elliptical plate model[30]. The estimated $E_{Y}$ is within the range of
reported values for bulk NbSe2 flakes[43, 44]. These quantities are obtained
without inducing damage on the flake, and are independent of the actuation
scheme.

Eq. (4) does not explain the asymmetric sinusoidal waves propagating beyond
the drum edges seen in Figs. 4(e-f). These waves are signatures of support
losses due to imperfect flake clamping at the edges[45]. Discussing the waves’
origin goes beyond the scope of this study, though resolving the waves’
amplitude, which is $1/3$ of $z_{max}$, demonstrate the capability of our
method to visualize acoustic waves in NMRs [17].

In summary, we demonstrated an in-situ, non-invasive method of calibrating
motional displacement of plate NMRs by exploiting wave interference phenomena
in FP cavities. Using a probe laser beam, and applying MIA to different
realizations, we determine cross-sectional profiles of the NMR thickness and
spacer height, transduction factors of NbSe2 plate resonators, and
subnanometer motional amplitudes that helped examine modal properties of the
drumheads. We foresee that this method will be applicable to flexural and
acoustic wave resonators of various geometries.

###### Acknowledgements.

We acknowledge the contributions of T.-H. Hsu and W.-H. Chang in fabricating
devices and in building the experimental setup. We thank A.F. Rigosi for
sharing the measured dielectric constant spectra of bulk and few-layers of
NbSe2. We thank the Taiwan International Graduate Program for the financial
support. This project is funded by Academia Sinica Grand Challenge Seed
Program (AS-GC-109-08), Ministry of Science and Technology (MOST) of Taiwan
(107-2112-M-001-001-MY3), Cost Share Programme (107-2911-I-001-511), the Royal
Society International Exchanges Scheme (grant
IES$\backslash$R3$\backslash$170029), and iMATE(2391-107-3001). We extend our
gratitude for the Academia Sinica Nanocore Facility. Attributions.-M.A.C.A.
and J.C.E. contributed equally in this work. C.-D.C. conceived the devices and
supervised the project; J.C.E. fabricated the devices. M.A.C.A. and J.-Y.W.
modeled the calibration scheme. C.-Y.Y. and K.-H.L. designed and built the
setup for optical measurements. M.A.C.A., J.C.E. and C.-Y.Y. performed the
experiment. M.A.C.A., J.C.E., J.-Y.W., T.-H.L., K.-S.C.-L., S.K., Y.P. and
C.-D.C. analyzed the data, performed simulations and wrote the manuscript; all
authors discussed the results and contributed to the manuscript.

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