arXiv subjects

Armin Eftekhari

Publications and source records attributed to Armin Eftekhari.

36 records · Page 2Linked to original sources

arXiv2019

Nearly Minimal Over-Parametrization of Shallow Neural Networks

A recent line of work has shown that an overparametrized neural network can perfectly fit the training data, an otherwise often intractable nonconvex optimization problem. For (fully-connected) shallow networks, in the best case scenario, the existing theory requires quadratic over-parametrization as a function of the number of training samples. This paper establishes that linear overparametrization is sufficient to fit the training data, using a simple variant of the (stochastic) gradient descent. Crucially, unlike several related works, the training considered in this paper is not limited to the lazy regime in the sense cautioned against in [1, 2]. Beyond shallow networks, the framework developed in this work for over-parametrization is applicable to a variety of learning problems.

cs.LG↗

arXiv2019

Randomized Learning of the Second-Moment Matrix of a Smooth Function

Consider an open set $\mathbb{D}\subseteq\mathbb{R}^n$, equipped with a probability measure $μ$. An important characteristic of a smooth function $f:\mathbb{D}\rightarrow\mathbb{R}$ is its \emph{second-moment matrix} $Σ_μ:=\int \nabla f(x) \nabla f(x)^* μ(dx) \in\mathbb{R}^{n\times n}$, where $\nabla f(x)\in\mathbb{R}^n$ is the gradient of $f(\cdot)$ at $x\in\mathbb{D}$ and $*$ stands for transpose. For instance, the span of the leading $r$ eigenvectors of $Σ_μ$ forms an \emph{active subspace} of $f(\cdot)$, which contains the directions along which $f(\cdot)$ changes the most and is of particular interest in \emph{ridge approximation}. In this work, we propose a simple algorithm for estimating $Σ_μ$ from random point evaluations of $f(\cdot)$ \emph{without} imposing any structural assumptions on $Σ_μ$. Theoretical guarantees for this algorithm are established with the aid of the same technical tools that have proved valuable in the context of covariance matrix estimation from partial measurements.

cs.IT↗

arXiv2019

Fast and Provable ADMM for Learning with Generative Priors

In this work, we propose a (linearized) Alternating Direction Method-of-Multipliers (ADMM) algorithm for minimizing a convex function subject to a nonconvex constraint. We focus on the special case where such constraint arises from the specification that a variable should lie in the range of a neural network. This is motivated by recent successful applications of Generative Adversarial Networks (GANs) in tasks like compressive sensing, denoising and robustness against adversarial examples. The derived rates for our algorithm are characterized in terms of certain geometric properties of the generator network, which we show hold for feedforward architectures, under mild assumptions. Unlike gradient descent (GD), it can efficiently handle non-smooth objectives as well as exploit efficient partial minimization procedures, thus being faster in many practical scenarios.

cs.LG↗

arXiv2019

Sparse Inverse Problems Over Measures: Equivalence of the Conditional Gradient and Exchange Methods

We study an optimization program over nonnegative Borel measures that encourages sparsity in its solution. Efficient solvers for this program are in increasing demand, as it arises when learning from data generated by a `continuum-of-subspaces' model, a recent trend with applications in signal processing, machine learning, and high-dimensional statistics. We prove that the conditional gradient method (CGM) applied to this infinite-dimensional program, as proposed recently in the literature, is equivalent to the exchange method (EM) applied to its Lagrangian dual, which is a semi-infinite program. In doing so, we formally connect such infinite-dimensional programs to the well-established field of semi-infinite programming. On the one hand, the equivalence established in this paper allows us to provide a rate of convergence for EM which is more general than those existing in the literature. On the other hand, this connection and the resulting geometric insights might in the future lead to the design of improved variants of CGM for infinite-dimensional programs, which has been an active research topic. CGM is also known as the Frank-Wolfe algorithm.

math.OC↗

arXiv2018

Streaming Principal Component Analysis From Incomplete Data

Linear subspace models are pervasive in computational sciences and particularly used for large datasets which are often incomplete due to privacy issues or sampling constraints. Therefore, a critical problem is developing an efficient algorithm for detecting low-dimensional linear structure from incomplete data efficiently, in terms of both computational complexity and storage. In this paper we propose a streaming subspace estimation algorithm called Subspace Navigation via Interpolation from Partial Entries (SNIPE) that efficiently processes blocks of incomplete data to estimate the underlying subspace model. In every iteration, SNIPE finds the subspace that best fits the new data block but remains close to the previous estimate. We show that SNIPE is a streaming solver for the underlying nonconvex matrix completion problem, that it converges globally to a stationary point of this program regardless of initialization, and that the convergence is locally linear with high probability. We also find that SNIPE shows state-of-the-art performance in our numerical simulations.

cs.IT↗

arXiv2018

Weighted Matrix Completion and Recovery with Prior Subspace Information

An incoherent low-rank matrix can be efficiently reconstructed after observing a few of its entries at random, and then solving a convex program that minimizes the nuclear norm. In many applications, in addition to these entries, potentially valuable prior knowledge about the column and row spaces of the matrix is also available to the practitioner. In this paper, we incorporate this prior knowledge in matrix completion---by minimizing a weighted nuclear norm---and precisely quantify any improvements. In particular, we find in theory that reliable prior knowledge reduces the sample complexity of matrix completion by a logarithmic factor, and the observed improvement in numerical simulations is considerably more magnified. We also present similar results for the closely related problem of matrix recovery from generic linear measurements.

cs.IT↗

arXiv2018

PCA by Determinant Optimization has no Spurious Local Optima

Principal component analysis (PCA) is an indispensable tool in many learning tasks that finds the best linear representation for data. Classically, principal components of a dataset are interpreted as the directions that preserve most of its "energy", an interpretation that is theoretically underpinned by the celebrated Eckart-Young-Mirsky Theorem. There are yet other ways of interpreting PCA that are rarely exploited in practice, largely because it is not known how to reliably solve the corresponding non-convex optimisation programs. In this paper, we consider one such interpretation of principal components as the directions that preserve most of the "volume" of the dataset. Our main contribution is a theorem that shows that the corresponding non-convex program has no spurious local optima. We apply a number of solvers for empirical confirmation.

math.OC↗

arXiv2017

MC^2: A Two-Phase Algorithm for Leveraged Matrix Completion

Leverage scores, loosely speaking, reflect the importance of the rows and columns of a matrix. Ideally, given the leverage scores of a rank-$r$ matrix $M\in\mathbb{R}^{n\times n}$, that matrix can be reliably completed from just $O(rn\log^{2}n)$ samples if the samples are chosen randomly from a nonuniform distribution induced by the leverage scores. In practice, however, the leverage scores are often unknown a priori. As such, the sample complexity in uniform matrix completion---using uniform random sampling---increases to $O(η(M)\cdot rn\log^{2}n)$, where $η(M)$ is the largest leverage score of $M$. In this paper, we propose a two-phase algorithm called MC$^2$ for matrix completion: in the first phase, the leverage scores are estimated based on uniform random samples, and then in the second phase the matrix is resampled nonuniformly based on the estimated leverage scores and then completed. For well-conditioned matrices, the total sample complexity of MC$^2$ is no worse than uniform matrix completion, and for certain classes of well-conditioned matrices---namely, reasonably coherent matrices whose leverage scores exhibit mild decay---MC$^2$ requires substantially fewer samples. Numerical simulations suggest that the algorithm outperforms uniform matrix completion in a broad class of matrices, and in particular, is much less sensitive to the condition number than our theory currently requires.

cs.IT↗

arXiv2017

Stabilizing Embedology: Geometry-Preserving Delay-Coordinate Maps

Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens' embedding theorem, which guarantees that delay-coordinate maps use the time-series output to provide a reconstruction of the hidden state space that is a one-to-one embedding of the system's attractor. While this topological guarantee ensures that distinct points in the reconstruction correspond to distinct points in the original state space, it does not characterize the quality of this embedding or illuminate how the specific parameters affect the reconstruction. In this paper, we extend Takens' result by establishing conditions under which delay-coordinate mapping is guaranteed to provide a stable embedding of a system's attractor. Beyond only preserving the attractor topology, a stable embedding preserves the attractor geometry by ensuring that distances between points in the state space are approximately preserved. In particular, we find that delay-coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the attractor. The stable rank reflects the relation between the sampling interval and the number of delays in delay-coordinate mapping. Our theoretical findings give guidance to choosing system parameters, echoing the trade-off between irrelevancy and redundancy that has been heuristically investigated in the literature. Our initial result is stated for attractors that are smooth submanifolds of Euclidean space, with extensions provided for the case of strange attractors.

nlin.CD↗

arXiv2017

A near-stationary subspace for ridge approximation

Response surfaces are common surrogates for expensive computer simulations in engineering analysis. However, the cost of fitting an accurate response surface increases exponentially as the number of model inputs increases, which leaves response surface construction intractable for high-dimensional, nonlinear models. We describe ridge approximation for fitting response surfaces in several variables. A ridge function is constant along several directions in its domain, so fitting occurs on the coordinates of a low-dimensional subspace of the input space. We review essential theory for ridge approximation---e.g., the best mean-squared approximation and an optimal low-dimensional subspace---and we prove that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. Motivated by the theory, we propose a computational heuristic that uses an estimated active subspace as an initial guess for a ridge approximation fitting problem. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. We then propose a simple alternating heuristic for fitting a ridge function, and we demonstrate the effectiveness of the active subspace initial guess applied to an airfoil model of drag as a function of its 18 shape parameters.

math.NA↗

arXiv2017

What to Expect When You Are Expecting on the Grassmannian

Consider an incoming sequence of vectors, all belonging to an unknown subspace $\operatorname{S}$, and each with many missing entries. In order to estimate $\operatorname{S}$, it is common to partition the data into blocks and iteratively update the estimate of $\operatorname{S}$ with each new incoming measurement block. In this paper, we investigate a rather basic question: Is it possible to identify $\operatorname{S}$ by averaging the column span of the partially observed incoming measurement blocks on the Grassmannian? We show that in general the span of the incoming blocks is in fact a biased estimator of $\operatorname{S}$ when data suffers from erasures, and we find an upper bound for this bias. We reach this conclusion by examining the defining optimization program for the Fréchet expectation on the Grassmannian, and with the aid of a sharp perturbation bound and standard large deviation results.

cs.IT↗

arXiv2016

What Happens to a Manifold Under a Bi-Lipschitz Map?

We study geometric and topological properties of the image of a smooth submanifold of $\mathbb{R}^{n}$ under a bi-Lipschitz map to $\mathbb{R}^{m}$. In particular, we characterize how the dimension, diameter, volume, and reach of the embedded manifold relate to the original. Our main result establishes a lower bound on the reach of the embedded manifold in the case where $m \le n$ and the bi-Lipschitz map is linear. We discuss implications of this work in signal processing and machine learning, where bi-Lipschitz maps on low-dimensional manifolds have been constructed using randomized linear operators.

cs.IT↗

arXiv2015

Greed is Super: A Fast Algorithm for Super-Resolution

We present a fast two-phase algorithm for super-resolution with strong theoretical guarantees. Given the low-frequency part of the spectrum of a sequence of impulses, Phase I consists of a greedy algorithm that roughly estimates the impulse positions. These estimates are then refined by local optimization in Phase II. In contrast to the convex relaxation proposed by Candès et al., our approach has a low computational complexity but requires the impulses to be separated by an additional logarithmic factor to succeed. The backbone of our work is the fundamental work of Slepian et al. involving discrete prolate spheroidal wave functions and their unique properties.

cs.IT↗

arXiv2015

Computing Active Subspaces Efficiently with Gradient Sketching

Active subspaces are an emerging set of tools for identifying and exploiting the most important directions in the space of a computer simulation's input parameters; these directions depend on the simulation's quantity of interest, which we treat as a function from inputs to outputs. To identify a function's active subspace, one must compute the eigenpairs of a matrix derived from the function's gradient, which presents challenges when the gradient is not available as a subroutine. We numerically study two methods for estimating the necessary eigenpairs using only linear measurements of the function's gradient. In practice, these measurements can be estimated by finite differences using only two function evaluations, regardless of the dimension of the function's input space.

math.NA↗

arXiv2014

A First Analysis of the Stability of Takens' Embedding

Takens' Embedding Theorem asserts that when the states of a hidden dynamical system are confined to a low-dimensional attractor, complete information about the states can be preserved in the observed time-series output through the delay coordinate map. However, the conditions for the theorem to hold ignore the effects of noise and time-series analysis in practice requires a careful empirical determination of the sampling time and number of delays resulting in a number of delay coordinates larger than the minimum prescribed by Takens' theorem. In this paper, we use tools and ideas in Compressed Sensing to provide a first theoretical justification for the choice of the number of delays in noisy conditions. In particular, we show that under certain conditions on the dynamical system, measurement function, number of delays and sampling time, the delay-coordinate map can be a stable embedding of the dynamical system's attractor.

math.DS↗

arXiv2014

New Analysis of Manifold Embeddings and Signal Recovery from Compressive Measurements

Compressive Sensing (CS) exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive, often random linear measurements of that signal. Strong theoretical guarantees have been established concerning the embedding of a sparse signal family under a random measurement operator and on the accuracy to which sparse signals can be recovered from noisy compressive measurements. In this paper, we address similar questions in the context of a different modeling framework. Instead of sparse models, we focus on the broad class of manifold models, which can arise in both parametric and non-parametric signal families. Using tools from the theory of empirical processes, we improve upon previous results concerning the embedding of low-dimensional manifolds under random measurement operators. We also establish both deterministic and probabilistic instance-optimal bounds in $\ell_2$ for manifold-based signal recovery and parameter estimation from noisy compressive measurements. In line with analogous results for sparsity-based CS, we conclude that much stronger bounds are possible in the probabilistic setting. Our work supports the growing evidence that manifold-based models can be used with high accuracy in compressive signal processing.

cs.IT↗

arXiv2014

The Restricted Isometry Property for Random Block Diagonal Matrices

In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. It is by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIP under certain conditions on the number of measurements. Their use can be limited in practice, however, due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. These issues have recently motivated considerable effort towards studying the RIP for structured random matrices. In this paper, we study the RIP for block diagonal measurement matrices where each block on the main diagonal is itself a sub-Gaussian random matrix. Our main result states that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on certain properties of the basis in which the signals are sparse. In the best case, these matrices perform nearly as well as dense Gaussian random matrices, despite having many fewer nonzero entries.

cs.IT↗

arXiv2012

Matched Filtering from Limited Frequency Samples

In this paper, we study a simple correlation-based strategy for estimating the unknown delay and amplitude of a signal based on a small number of noisy, randomly chosen frequency-domain samples. We model the output of this "compressive matched filter" as a random process whose mean equals the scaled, shifted autocorrelation function of the template signal. Using tools from the theory of empirical processes, we prove that the expected maximum deviation of this process from its mean decreases sharply as the number of measurements increases, and we also derive a probabilistic tail bound on the maximum deviation. Putting all of this together, we bound the minimum number of measurements required to guarantee that the empirical maximum of this random process occurs sufficiently close to the true peak of its mean function. We conclude that for broad classes of signals, this compressive matched filter will successfully estimate the unknown delay (with high probability, and within a prescribed tolerance) using a number of random frequency-domain samples that scales inversely with the signal-to-noise ratio and only logarithmically in the in the observation bandwidth and the possible range of delays.

cs.IT↗