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Title: Spectral Inference Networks: Unifying Deep and Spectral Learning

Paper content:
We present Spectral Inference Networks, a framework for learning eigenfunctions of linear operators by stochastic optimization.
Spectral Inference Networks generalize Slow Feature Analysis to generic symmetric operators, and are closely related to Variational Monte Carlo methods from computational physics.
As such, they can be a powerful tool for unsupervised representation learning from video or graph-structured data.
We cast training Spectral Inference Networks as a bilevel optimization problem, which allows for online learning of multiple eigenfunctions.
We show results of training Spectral Inference Networks on problems in quantum mechanics and feature learning for videos on synthetic datasets.
Our results demonstrate that Spectral Inference Networks accurately recover eigenfunctions of linear operators and can discover interpretable representations from video in a fully unsupervised manner.
Spectral algorithms are central to machine learning and scientific computing.
In machine learning, eigendecomposition and singular value decomposition are foundational tools, used for PCA as well as a wide variety of other models.
In scientific applications, solving for the eigenfunction of a given linear operator is central to the study of PDEs, and gives the time-independent behavior of classical and quantum systems.
For systems where the linear operator of interest can be represented as a reasonably-sized matrix, full eigendecomposition can be achieved in O(n 3 ) time BID11 , and in cases where the matrix is too large to diagonalize completely (or even store in memory), iterative algorithms based on Krylov subspace methods can efficiently compute a fixed number of eigenvectors by repeated application of matrix-vector products (Golub & Van Loan, 2012) .At
a larger scale, the eigenvectors themselves cannot be represented explicitly in memory. This
is the case in many applications in quantum physics and machine learning, where the state space of interest may be combinatorially large or even continuous and high dimensional. Typically
, the eigenfunctions of interest are approximated from a fixed number of points small enough to be stored in memory, and then the value of the eigenfunction at other points is approximated by use of the Nyström method (Bengio et al., 2004) . As this
depends on evaluating a kernel between a new point and every point in the training set, this is not practical for large datasets, and some form of function approximation is necessary. By choosing
a function approximator known to work well in a certain domain, such as convolutional neural networks for vision, we may be able to bias the learned representation towards reasonable solutions in a way that is difficult to encode by choice of kernel.In this paper, we propose a way to approximate eigenfunctions of linear operators on highdimensional function spaces with neural networks, which we call Spectral Inference Networks (SpIN). We show how
to train these networks via bilevel stochastic optimization. Our method
finds correct eigenfunctions of problems in quantum physics and discovers interpretable representations from video. This significantly
extends prior work on unsupervised learning without a generative model and we expect will be useful in scaling many applications of spectral methods.The outline of the paper is as follows. Sec 2 provides a review
of related work on spectral learning and stochastic optimization of approximate eigenfunctions. Sec. 3 defines the objective
function for Spectral Inference Networks, framing eigenfunction problems as an optimization problem. Sec. 4 describes the algorithm
for training Spectral Inference Networks using bilevel optimization and a custom gradient to learn ordered eigenfunctions simultaneously. Experiments are presented in Sec.
5 and future directions are discussed in Sec. 6. We also include supplementary materials
with more in-depth derivation of the custom gradient updates (Sec. A), a TensorFlow implementation of the core algorithm (Sec. B), and additional experimental results and training details (Sec. C).
We have shown that a single unified framework is able to compute spectral decompositions by stochastic gradient descent on domains relevant to physics and machine learning.
This makes it possible to learn eigenfunctions over very high-dimensional spaces from very large datasets and generalize to new data without the Nyström approximation.
This extends work using slowness as a criterion for unsupervised learning without a generative model, and addresses an unresolved issue with biased gradients due to finite batch size.
A limitation of the proposed solution is the requirement of computing full Jacobians at every time step, and improving the scaling of training is a promising direction for future research.
The physics application presented here is on a fairly simple system, and we hope that Spectral Inference Nets can be fruitfully applied to more complex physical systems for which computational solutions are not yet available.
The representations learned on video data show nontrivial structure and sensitivity to meaningful properties of the scene.
These representations could be used for many downstream tasks, such as object tracking, gesture recognition, or faster exploration and subgoal discovery in reinforcement learning.
Finally, while the framework presented here is quite general, the examples shown investigated only a small number of linear operators.
Now that the basic framework has been laid out, there is a rich space of possible kernels and architectures to combine and explore.

assistant: text: This paper presents a framework to learn eigenfunctions via a stochastic process and proposes to tackle the challenge of computing eigenfunctions in a large-scale context by approximating then using a two-phase stochastic optimization process.