Shining light on data

Mohan Sarovar
Sandia National Labs
Livermore, CA 94550
mnsarov@sandia.gov

Akshat Kumar
Department of Mathematics
Clarkson University
Potsdam, NY 13699
akumar@clarkson.edu

Abstract
Experimental sciences have come to depend heavily on our ability to organize,interpret and analyze high-dimensional datasets produced from observations of alarge number of variables governed by natural processes. Natural laws, conservation
principles, and dynamical structure introduce intricate inter-dependencies among
these observed variables, which in turn yield geometric structure, with fewer
degrees of freedom, on the dataset. We show how fine-scale features of this structure
in data can be extracted from discrete approximations to quantum mechanicalprocesses given by data-driven graph Laplacians and localized wavepackets. This
leads to a novel, yet natural uncertainty principle for data analysis induced bylimited data. We illustrate some applications to learning with algorithms on several
model examples and real-world datasets.
Nature is complex, yet organized – this basic facet has become a cornerstone for how we analyze
and interpret vast amounts of data ranging from the biological, physical, geological, meteorological,
all the way to the astronomical. In fact, we owe to this, nearly all of the significant advances indata analysis disciplines in the past two decades, including signals processing [8, 22] and machinelearning [3, 5, 11], where tractable models capture, interpret and reproduce complex natural data. We
can turn this observation on its head and ask: are there certain natural processes that are fundamental
to understanding the structure in complex, yet organized data?
We propose that, much like how quantum mechanics models nature at fine scales, the fine-scaled
resolution of the organization and structure of data is also best characterized using quantum mechanical processes. Consider the following general scenario: an experimentalist makes a sequence ofmeasurements, each consisting of a value for D variables. Each measurement is thus a point in RD .
Of course, measurements coming from nature are bound by physical laws, so in fact one must imagine
that the true ambient space is some other, nonlinear structure M, residing in RD . Moreover, if theexperiment is governed by a number of parameters, then these give M its local degrees of freedom
around any given measurement. This principle is in fact often studied and known as the manifold
hypothesis (MH) [9]: measurements of arbitrarily high dimensions (residing in RD ) arising fromnatural processes are confined to low-dimensional manifolds (i.e., M ⊂ RD with dim M ≪ D).
Due to the MH, the experimentalist’s aim of connecting model parameters to observations through therelationships among measurements is just the analysis of the organizational structure, or geometry ofdata. In the natural sciences, such analyses are commonly performed through dimensionality reduction, classification, etc. using techniques such as principal component analysis (PCA), t-distributed
stochastic neighbor embedding (t-SNE), or variants of Laplacian eigenmaps, that indirectly probe thegeometry of data, e.g., [7, 19, 14]. In addition, it has recently been appreciated that understanding andexploiting the structure of data, and sometimes reorganizing/reparameterizing it, has advantages inlearning frameworks such as convolutional neural networks, e.g., [24, 4]. While the state-of-the-art inlearning the structure and organization of data is founded on principles of Markovian dynamics [2, 6]– primarily because of their local nature that forms an accessible link between random walks andMarkov processes – in practice, these are suitable only for accessing coarse features of the data due toMachine Learning and the Physical Sciences workshop, NeurIPS 2022.

the slowly varying nature of the steady-states that form the basis of these techniques. We propose thatto perform a fine-scaled data analysis, it is advantageous to shift the paradigm to quantum mechanics,
which provides a natural way to formulate dynamics on the data that respects limits to resolution ofits geometric structure set by data sampling, through the introduction of phase-space uncertainty.
1: Inputs: XN = {v1 , ..., vN }, v ∗ , ϵ > 0, α ≥ 1, t > 0
2: Output: Propagated state [ψhζ ](t)3: procedure P ROPAGATE
2
4:
Compute [Tϵ ]i,j = k(||vi −vj || /ϵ)P
5:Compute diagonal matrix [Dϵ ]i,i = N
j=1 [Tϵ ]i,j

Algorithm 1: Pseudocode for algorithm

that performs data-driven propagation of coherent states on a data graph. The inputs are thedataset XN ⊂ RD , an initial data point topropagate from, v ∗ , parameters ϵ > 0, α ≥ 1,
and a time to propagate for, t > 0. Lines
4-7 construct a data-driven quantum propagat4(IN −Dϵ−1 Tϵ )tor, Uϵ,Nusing a graph Laplacian ∆ϵ,N comCompute ∆ϵ,N =ϵpputed from the data. k(·) (line 4) is an ext
Uϵ,N ← exp(−it ∆ϵ,N )ponentially decaying function of the argument
1and IN (line 6) is the N × N identify maSet h = ϵ (2+α)
trix. Lines 10-11 form the N × 1 vector thatSet p0 = vj −v ∗ for vj closest to point v ∗ ⇒ ζ = (v ∗ , p0/||p0 ||) approximates a coherent state at phase space
while 1 ≤ ℓ ≤ N do
point ζ := (v ∗ , p0/||p0 ||). A more detailedi (v −v ∗ )T p /||p ||ζ−||vℓ −v∗ ||2/2h h
description of this algorithm, and extensions of0ℓ0[ψh ]ℓ ← eeit, are presented in [17, 16].

6:
7:
8:
9:
10:
11:
12:

t
return [ψhζ ](t) = Uϵ,N[ψhζ ]We realize this proposal through the simulation of quantum dynamics on data as sketched in Algorithm
1. The fundamental relationship connecting this dynamics to the structure of data is given by a discrete
quantum-classical correspondence principle (QCC), which is established in Ref. [15] and summarizedin [17, 16] and applies when the MH holds for the data in RD , confined to M a smooth, compact,boundaryless submanifold. This relationship is supported by the traditional form of the QCC,
which physically in our geometric setting connects the propagation of wavefunctions of photons(quantized excitations) on curved space (M) with geodesic propagation of light rays [12]. Such a
propagation is illustrated in green in Figure 2. More formally, it is a well-known result in microlocal
analysis [23, 25]√that the trajectory of an impulse δx∗ at x∗ ∈ M with respect to the propagation
U t [δx∗ ] := e−it ∆ [δx∗ ], where ∆ is the Laplace-Beltrami operator of M, has singular support
along geodesics in all directions at distance |t| < T from x∗ , for a bounded time T . This is thetcontinuum QCC√ in action: it connects the quantum mechanical propagation U [δx∗ ] generated by thequantization ∆ of the classical Hamiltonian (purely kinetic energy), to the geodesic flow, by moving
the energy concentration of the singular state δx∗ to time t along the classical flow in directions of
the state’s asymptotically large momenta (i.e., the frequencies of its wavefunction) [23].
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0

/2

3 /2

p=1

2

position

(a)

(b)

(c)

Figure 2: (a) Data is N = 2500 regularly spaced samples from the unit circle: xj = (cos θj , sin θj ) for θj ∈ [0, 2π). (b) We observet2
the squared magnitude of: (green) an undirected optical ray from x∗ = π at time t = π
2 , i.e., |U [δπ ]| , and (blue) the correspondingt
data-driven quantum propagation, i.e., |Uϵ,N[δπ ]|2 . While the former is concentrated at x− = π − t = −π/2, x+ = π + t = 3π/2,
which are points that minimize |dg (x∗ , x) − t| (dg is geodesic distance), the latter state bears no resemblance to geodesic propagation, withsignificant attenuation over the green signal, and an undiminished peak at the source, x∗ . We also show (dashed red) the data-driven propagation
ζ 2
tof a coherent state centered at x∗ with uncertainty parameter h, i.e., |Uϵ,N[ψh]| , with ζ = (π, 1). This state is approximately centered at
the point x+ , and in fact, its expected position, ⟨x⟩, satisfies |⟨x⟩ − x+ | ≤ h. Note that all curves are normalized to have the same maximum
value in order to√plot them on the same scale. (c) Depiction of coherent state propagation in phase space, T ∗ M. The zoom-in schematically√shows ϵ and h, the data-determined scale and uncertainty parameters, respectively.

Coming back to quantum dynamics on the data, we couple the above continuum QCC with the
now well-known result that, assuming the MH, we can approximate from N measurement samples
2

XN := {v1 , . . . , vN } ⊂ RD theof their underlying manifold through√ Laplace-Beltrami operator√
the graph Laplacian ∆ϵ,N of a ϵ-nearest-neighbour ( ϵ-n.n.) graph on XN with high probability(w.h.p.) [2, 13, 6]. Our recent work shows that a data-driven approximation
to the quantum mechanical√−it ∆ϵ,Nttpropagator U is given w.h.p., by the N × N matrix Uϵ,N := e
. Given this construction, itt
is tempting to reproduce the geodesic propagation of light rays given through U t [δx∗ ] with Uϵ,N[δx∗ ].
However, as shown in the blue curve in Figure 2(b), this data-driven propagation bears no resemblancetto the continuum signal√U [δx∗ ] (green curve). This is because ∆ϵ,N is defined through a Markov
process discretized to ϵ-balls on the manifold and based on the uncertainty principle, our recent
work shows that thiswith
√ Markov process acts like a low-pass filter by attenuating spatial frequencies√
magnitude ≳ 1/ ϵ. Therefore, ∆ϵ,N approximates a scalar operator at bandwidth ≳ 1/ ϵ and bytvirtue of being its spectral function, so does Uϵ,N
. This explains the concentration of the blue curve∗in Figure 2(b) about x and predicts that this behaviour will persist even in the limit N → ∞ while
ϵ > 0 remains bounded away from zero.
Understanding this issue is the fulcrum of our discrete QCC, which recovers light rays from datadriven simulation of quantum dynamics. As we’ve seen, the primary obstruction to approximatingt
the continuum QCC with Uϵ,N[δx∗ ] is the over-concentration of the initial state δx∗ , which gives√
frequency content above the data-determined threshold ∼ 1/ ϵ. To control this, we introduce a stateψh ∈ C ∞ (M × (0, 1]h ) surrogate to δx∗ , whose bandwidth we can control to scale as ∼ 1/h and
t
∗
whose
√ continuum propagation U [ψh ] follows the light ray emanating from x until time t, to withina h-ball throughout the propagation. This is satisfied by coherent states, which in the continuum
∗ 2
∗
1
i
have position-space representation: ψhζ (v) := e− 2h ||v−x || e− h ⟨x −v,p⟩ with position x∗ ∈ M andmomentum p ∈ Tx∗∗ M. Rooted in the above arguments, it is established in Ref. [15] and furthertdiscussed in [17, 16] that in order for the data-driven propagation Uϵ,N[ψhζ ] to approximate U t [ψhζ ]√(w.h.p.) and therefore, to follow the light rays on data, the bandwidth 1/h must be kept ≪ 1/ ϵ.
Thus, the uncertainty parameter 0 < h ≤ 1 governing the quantum mechanical properties of coherent
states gives us the desired control to establish our discrete QCC.
In Figure 2(b) the dashed red line shows the result of propagating a coherent state using a datadriven propagator for the circle example. By incorporating the intrinsic uncertainty induced by finite
sampling into the formulation of quantum dynamics, we recover accurate propagation. In Figure 2(c)we also depict coherent state propagation in phase space for this example, and graphically show the
relationship between ϵ and h.
In [15, 17, 16], convergence results have been derived to establish the following: given N data
samples, XN , from a smooth density on M, the data-driven finite-dimensional matrix propagation1t[ψhζ ](t) := Uϵ,N[ψhζ ] returned by Algorithm 1, with h ∝ ϵ 2+α for α ≥ 1 agrees with U t [ψhζ ] up
1

to uniform error O(h) w.h.p., provided h ≳ √N − γ for γ > 0 a constant depending only on α andζ
dim M. Thus, [ψh ](t) traverses within an O( h) radius of the geodesic beam emanating from v ∗ inPN
the direction p to time t. Moreover, the point x̄t := ||ψhζ (t)||−2 j=1 ( vj,1 ··· vj,D )|[ψhζ ](t)|2 , whichis the expected position of the propagated state, is w.h.p. within geodesic distance O(h) to the pointx∗t , that is geodesic distance t from the initial position v ∗ in the direction p. While this expectedvalue of the position coordinate, x̄t , is the best estimate of position along the geodesic path, due tothe localization of the propagated state the maximum of the wavepacket
distribution, we show that√x̂t := arg max |[ψhζ ](t)|2 is w.h.p. within geodesic distance O( h) from x∗t .
The data-driven quantum dynamics formulated above enables estimation of intrinsic distances between
points in a dataset: if x̄t (j; pj ) ∈ XN denotes the data point closest to x̄t (or x̂t ) computed as above
with respect to an initial state ψhζ localized at ζ := (vj , pj ), then t gives the propagation time of acoherent state following approximately a ray emanating in direction pj from vj . When the data issampled from a smooth density on a smooth, compact, boundaryless manifold, we have establishedthat t is w.h.p., within O(h) of the geodesic distance. In fact, by using local PCA to define ψhζ andhence, pj , it is shown in [17, 16] that the dataset XN can be charted with geodesic polar coordinates(GPC), or equivalently, normal coordinates. We pause to emphasize that the procedure we havedescribed gives access to geodesics and GPC on a manifold, which are inherently described by
non-linear dynamical equations, through linear, matrix computations. Computing such quantities,
even when much more is known about the manifold, is generally computationally difficult since
typical approximation methods are insufficient: polyhedral approximations are not a faithful model3

Avg. SAH fraction

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Apr

May

date

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Jun
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(b)
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Figure 3: Analysis of adherence to social distancing measures during the COVID-19 pandemic using the Social Distancing Metric dataset
from SafeGraph Inc. [1]. This dataset is a collection of geolocation information from mobile devices in the United States, aggregated at thecensus block group (CBG) level and recorded daily for a period of over a year. As previously studied in Ref. [18], we compute the stay-at-home(SAH) fraction as a simple metric of adherence to social distancing from this data: for a given date, this is a measure of how curtailed mobility
was within a CBG. To compare with Ref. [18] we consider data for the state of Georgia (GA) and limit it to the 117-day time period fromFebruary 23, 2020 to June 19, 2020, which provides a snapshot of mobility patterns during the first three months of the pandemic. Afterremoving 17 CBGs with poor quality data, there are 5509 CBGs within the state. Therefore, our dataset XN has N = 5509 samples, each of
dimension 117. Top (a,b,c): Embedding of XN and clustering when the expected position x̄t is used to define points a geodesic distance t
away. Bottom (d,e,f): Embedding and clustering when the maximum x̂t is used to define points a geodesic distance t away. (a) and (d) show
3-dimensional embeddings of XN and clustering of the resulting points into five clusters using k-means clustering. (b) and (e) show the average
SAH fraction time series for each cluster. In (b) we see a clear separation of clusters by their SAH behavior, and in (e) we have identified ananomalous SAH pattern (in purple) in one of the clusters. (c) and (f) color code the CBGs in GA according to the cluster they belong to. (c)shows a clear rural-urban divide in degree of social distancing behavior, and (f) is a magnification of the Atlanta metropolitan region, because
many of the outlier CBGs identified are located in this area. We refer to [17, 16] for a detailed analysis, including parameter values and choiceheuristics.

of manifolds with curvature restrictions [20] and even forward marching type approximations are
known to be prone to failures [21]. Furthermore, our result establishes the first general convergence
result for geodesics and GPC from data.
More broadly, the assignment of a data point x̄t (j, pj ) = vk to a given data point vj is, in itself,
independent of further structural assumptions on the dataset. Even when the data XN is not guaranteed
to be sampled from a manifold, this defines a distance relationship between two points based ont
the quantum walk Uϵ,N[ψhζ ]. Based on this distance relationship we can build an N × N adjacency
matrix, G, for a graph XN on XN , with elements Gj,k = Gk,j = t. Repeating this process fora collection of initial points vj and time-steps, t1 , . . . , tm populates this adjacency matrix, which
captures a notion of distance between the data points in XN given by the quantum propagation
times of coherent states. We can perform emebeddings of XN to achieve tasks such as recovering
reduced-dimensional coordinates, clustering, classification, etc. We find that even the most classicalembedding of XN in few dimensions, such as the Fruchterman-Reingold (FR) method of springs andelectrostatic forces [10], recovers salient features of complex datasets. For example, Figure 3 shows
examples of FR embedding into 3 dimensions followed by k-means clustering of a 117-dimensional
dataset related to population mobility during the initial months of the COVID-19 pandemic. The
embedding enabled by our discrete QCC and subsequent clustering is able to meaningfully separate
geographic regions with differing levels of mobility as well as identify outlier patterns. On the samedataset, a state-of-the-art Markov-based approach was deployed in Ref.[18] and relied on analysis in
14 dimensions to identify poorer quality clusters (see Fig. 2 of Ref. [18]), and was also unable toidentify the outlier patterns in mobility.
We have argued that, similar to the Fourier transform being fundamental for signal analysis, quantum
propagation and sensing are fundamental for structure analysis in sampled data. The structure is
carried by the dynamics of rays emitted on the dataset, which are signals generated by the quantumtpropagation Uϵ,N[δx∗ ] that can be resolved with the aid of coherent states defined on the dataset.
When the data satisfies the MH with a compact, smooth, boundaryless manifold, these signals reveal
4the underlying geodesic flow that governs the data’s organizational patterns. Our experiments indicatethat these signals efficiently reveal underlying patterns and anomalies in even sparse datasets.

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Broad impact
In this work we develop a new framework for understanding structure within large datasets. These
results will be useful for analyzing and extracting meaning from a broad range of datasets, withapplications ranging from the analysis of statistical and experimental data to image analysis andto analysis of sensor and computer simulation data. The unique capability we have developed, toefficiently identify geodesic distances between data points, could be useful for performing heretofore
infeasible analyses. As an example, we have shown in Figure 3 the capability of our methods todetect anomalous behavioral patterns in mobility during the COVID-19 pandemic, which surpasses
the comparative state-of-the-art in literature; in the accompanying recent works, we have given
further examples of pattern recognition on a variety of real-world datasets. As such, this could havetremendous societal benefit, since efficient analysis of large datasets is a ubiquitous task in almost
every field of study. Negative societal impacts are also possible if the increased capabilities in data
analysis are utilized for purposes such as surveillance. As with any fundamental data analysis method,
the societal impact is eventually dictated by the way in which the method is applied.

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [No] We provide our assumptions
and experiments on data that meet those assumptions and some that don’t. Longer
discussion is in full-length works, while this is an extended abstract.
(c) Did you discuss any potential negative societal impacts of your work? [Yes](d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] We give only aflavor of our theoretical results and have included the necessary assumptions for them.
(b) Did you include complete proofs of all theoretical results? [No] This is an extended
abstract; the complete proofs are given in the corresponding research papers.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] We provided asample pseudocode and referred to the data we have used (in the real world example,
we’ve provided a citation with the URL). We are happy to provide code and datasetson request.
6

(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [No] These details are in the corresponding research papers and we are
happy to provide them on request as well.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No](d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [N/A]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes](b) Did you mention the license of the assets? [No](c) Did you include any new assets either in the supplemental material or as a URL? [No](d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [No](e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [No]5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable? [N/A](b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable? [N/A](c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation? [N/A]7