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Stephen Becker

Publications and source records attributed to Stephen Becker.

At least 55 records · Page 3Linked to original sources
arXiv2021

A stochastic subspace approach to gradient-free optimization in high dimensions

We present a stochastic descent algorithm for unconstrained optimization that is particularly efficient when the objective function is slow to evaluate and gradients are not easily obtained, as in some PDE-constrained optimization and machine learning problems. The algorithm maps the gradient onto a low-dimensional random subspace of dimension $\ell$ at each iteration, similar to coordinate descent but without restricting directional derivatives to be along the axes. Without requiring a full gradient, this mapping can be performed by computing $\ell$ directional derivatives (e.g., via forward-mode automatic differentiation). We give proofs for convergence in expectation under various convexity assumptions as well as probabilistic convergence results under strong-convexity. Our method extends the well-known Gaussian smoothing technique to descent in subspaces of dimension greater than one, opening the doors to new analysis of Gaussian smoothing when more than one directional derivative is used at each iteration. We also provide a finite-dimensional variant of a special case of the Johnson-Lindenstrauss lemma. Experimentally, we show that our method compares favorably to coordinate descent, Gaussian smoothing, gradient descent and BFGS (when gradients are calculated via forward-mode automatic differentiation) on problems from the machine learning and shape optimization literature.

math.OC↗
arXiv2021

A study of scalar optically-pumped magnetometers for use in magnetoencephalography without shielding

Scalar optically-pumped magnetometers (OPMs) are being developed in small packages with high sensitivities. The high common-mode rejection ratio of these sensors allows for detection of very small signals in the presence of large background fields making them ideally suited for brain imaging applications in unshielded environments. Despite a flurry of activity around the topic, questions remain concerning how well a dipolar source can be localized under such conditions, especially when using few sensors. In this paper, we investigate the source localization capabilities using an array of scalar OPMs in the presence of a large background field while varying dipole strength, sensor count, and forward model accuracy. We also consider localization performance as the orientation angle of the background field changes. Our results are validated experimentally through accurate localization using a phantom virtual array mimicking a current dipole in a conducting sphere in a large background field. Our results are intended to give researchers a general sense of the capabilities and limitations of scalar OPMs for magnetoencephalography systems.

physics.ins-det↗
arXiv2021

Approximate maximum likelihood estimators for linear regression with design matrix uncertainty

In this paper we consider regression problems subject to arbitrary noise in the operator or design matrix. This characterization appropriately models many physical phenomena with uncertainty in the regressors. Although the problem has been studied extensively for ordinary/total least squares, and via models that implicitly or explicitly assume Gaussianity, less attention has been paid to improving estimation for regression problems under general uncertainty in the design matrix. To address difficulties encountered when dealing with distributions of sums of random variables, we rely on the saddle point method to estimate densities and form an approximate log-likelihood to maximize. We show that the proposed method performs favorably against other classical methods.

stat.CO↗
arXiv2021

Stochastic Gradient Langevin Dynamics with Variance Reduction

Stochastic gradient Langevin dynamics (SGLD) has gained the attention of optimization researchers due to its global optimization properties. This paper proves an improved convergence property to local minimizers of nonconvex objective functions using SGLD accelerated by variance reductions. Moreover, we prove an ergodicity property of the SGLD scheme, which gives insights on its potential to find global minimizers of nonconvex objectives.

cs.LG↗
arXiv2020

Bounds for the tracking error of first-order online optimization methods

This paper investigates online algorithms for smooth time-varying optimization problems, focusing first on methods with constant step-size, momentum, and extrapolation-length. Assuming strong convexity, precise results for the tracking iterate error (the limit supremum of the norm of the difference between the optimal solution and the iterates) for online gradient descent are derived. The paper then considers a general first-order framework, where a universal lower bound on the tracking iterate error is established. Furthermore, a method using "long-steps" is proposed and shown to achieve the lower bound up to a fixed constant. This method is then compared with online gradient descent for specific examples. Finally, the paper analyzes the effect of regularization when the cost is not strongly convex. With regularization, it is possible to achieve a non-regret bound. The paper ends by testing the accelerated and regularized methods on synthetic time-varying least-squares and logistic regression problems, respectively.

math.OC↗
arXiv2020

Robust Least Squares for Quantized Data Matrices

In this paper we formulate and solve a robust least squares problem for a system of linear equations subject to quantization error in the data matrix. Ordinary least squares fails to consider uncertainty in the operator, modeling all noise in the observed signal. Total least squares accounts for uncertainty in the data matrix, but necessarily increases the condition number of the operator compared to ordinary least squares. Tikhonov regularization or ridge regression is frequently employed to combat ill-conditioning, but requires parameter tuning which presents a host of challenges and places strong assumptions on parameter prior distributions. The proposed method also requires selection of a parameter, but it can be chosen in a natural way, e.g., a matrix rounded to the 4th digit uses an uncertainty bounding parameter of 0.5e-4. We show here that our robust method is theoretically appropriate, tractable, and performs favorably against ordinary and total least squares.

math.OC↗
arXiv2020

Guarantees for the Kronecker Fast Johnson-Lindenstrauss Transform Using a Coherence and Sampling Argument

In the recent paper [Jin, Kolda & Ward, arXiv:1909.04801], it is proved that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is, in fact, a Johnson-Lindenstrauss transform, which had previously only been conjectured. In this paper, we provide an alternative proof of this, for when the KFJLT is applied to Kronecker vectors, using a coherence and sampling argument. Our proof yields a different bound on the embedding dimension, which can be combined with the bound in the paper by Jin et al. to get a better bound overall. As a stepping stone to proving our result, we also show that the KFJLT is a subspace embedding for matrices with columns that have Kronecker product structure. Lastly, we compare the KFJLT to four other sketch techniques in numerical experiments on both synthetic and real-world data.

math.NA↗
arXiv2020

Efficient Solvers for Sparse Subspace Clustering

Sparse subspace clustering (SSC) clusters $n$ points that lie near a union of low-dimensional subspaces. The SSC model expresses each point as a linear or affine combination of the other points, using either $\ell_1$ or $\ell_0$ regularization. Using $\ell_1$ regularization results in a convex problem but requires $O(n^2)$ storage, and is typically solved by the alternating direction method of multipliers which takes $O(n^3)$ flops. The $\ell_0$ model is non-convex but only needs memory linear in $n$, and is solved via orthogonal matching pursuit and cannot handle the case of affine subspaces. This paper shows that a proximal gradient framework can solve SSC, covering both $\ell_1$ and $\ell_0$ models, and both linear and affine constraints. For both $\ell_1$ and $\ell_0$, algorithms to compute the proximity operator in the presence of affine constraints have not been presented in the SSC literature, so we derive an exact and efficient algorithm that solves the $\ell_1$ case with just $O(n^2)$ flops. In the $\ell_0$ case, our algorithm retains the low-memory overhead, and is the first algorithm to solve the SSC-$\ell_0$ model with affine constraints. Experiments show our algorithms do not rely on sensitive regularization parameters, and they are less sensitive to sparsity misspecification and high noise.

cs.CV↗
arXiv2020

Locality-sensitive hashing in function spaces

We discuss the problem of performing similarity search over function spaces. To perform search over such spaces in a reasonable amount of time, we use {\it locality-sensitive hashing} (LSH). We present two methods that allow LSH functions on $\mathbb{R}^N$ to be extended to $L^p$ spaces: one using function approximation in an orthonormal basis, and another using (quasi-)Monte Carlo-style techniques. We use the presented hashing schemes to construct an LSH family for Wasserstein distance over one-dimensional, continuous probability distributions.

cs.LG↗
arXiv2020

Optimization and Learning with Information Streams: Time-varying Algorithms and Applications

There is a growing cross-disciplinary effort in the broad domain of optimization and learning with streams of data, applied to settings where traditional batch optimization techniques cannot produce solutions at time scales that match the inter-arrival times of the data points due to computational and/or communication bottlenecks. Special types of online algorithms can handle this situation, and this article focuses on such time-varying optimization algorithms, with emphasis on Machine Leaning and Signal Processing, as well as data-driven Control. Approaches for the design of time-varying or online first-order optimization methods are discussed, with emphasis on algorithms that can handle errors in the gradient, as may arise when the gradient is estimated. Insights on performance metrics and accompanying claims are provided, along with evidence of cases where algorithms that are provably convergent in batch optimization may perform poorly in an online regime. The role of distributed computation is discussed. Illustrative numerical examples for a number of applications of broad interest are provided to convey key ideas.

math.OC↗
arXiv2019

Improved Fixed-Rank Nyström Approximation via QR Decomposition: Practical and Theoretical Aspects

The Nystrom method is a popular technique that uses a small number of landmark points to compute a fixed-rank approximation of large kernel matrices that arise in machine learning problems. In practice, to ensure high quality approximations, the number of landmark points is chosen to be greater than the target rank. However, for simplicity the standard Nystrom method uses a sub-optimal procedure for rank reduction. In this paper, we examine the drawbacks of the standard Nystrom method in terms of poor performance and lack of theoretical guarantees. To address these issues, we present an efficient modification for generating improved fixed-rank Nystrom approximations. Theoretical analysis and numerical experiments are provided to demonstrate the advantages of the modified method over the standard Nystrom method. Overall, the aim of this paper is to convince researchers to use the modified method, as it has nearly identical computational complexity, is easy to code, has greatly improved accuracy in many cases, and is optimal in a sense that we make precise.

stat.ML↗
arXiv2019

One-Pass Sparsified Gaussian Mixtures

We present a one-pass sparsified Gaussian mixture model (SGMM). Given $N$ data points in $P$ dimensions, $X$, the model fits $K$ Gaussian distributions to $X$ and (softly) classifies each point to these clusters. After paying an up-front cost of $\mathcal{O}(NP\log P)$ to precondition the data, we subsample $Q$ entries of each data point and discard the full $P$-dimensional data. SGMM operates in $\mathcal{O}(KNQ)$ time per iteration for diagonal or spherical covariances, independent of $P$, while estimating the model parameters in the full $P$-dimensional space, making it one-pass and hence suitable for streaming data. We derive the maximum likelihood estimators for the parameters in the sparsified regime, demonstrate clustering on synthetic and real data, and show that SGMM is faster than GMM while preserving accuracy.

cs.LG↗
arXiv2019

Safe Feature Elimination for Non-Negativity Constrained Convex Optimization

Inspired by recent work on safe feature elimination for $1$-norm regularized least-squares, we develop strategies to eliminate features from convex optimization problems with non-negativity constraints. Our strategy is safe in the sense that it will only remove features/coordinates from the problem when they are guaranteed to be zero at a solution. To perform feature elimination we use an accurate, but not optimal, primal-dual feasible pair, making our methods robust and able to be used on ill-conditioned problems. We supplement our feature elimination problem with a method to construct an accurate dual feasible point from an accurate primal feasible point; this allows us to use a first-order method to find an accurate primal feasible point, then use that point to construct an accurate dual feasible point and perform feature elimination. Under reasonable conditions, our feature elimination strategy will eventually eliminate all zero features from the problem. As an application of our methods we show how safe feature elimination can be used to robustly certify the uniqueness of non-negative least-squares (NNLS) problems. We give numerical examples on a well-conditioned synthetic NNLS problem and a on set of 40000 extremely ill-conditioned NNLS problems arising in a microscopy application.

math.OC↗
arXiv2019

Resolvability of Hamming Graphs

A subset of vertices in a graph is called resolving when the geodesic distances to those vertices uniquely distinguish every vertex in the graph. Here, we characterize the resolvability of Hamming graphs in terms of a constrained linear system and deduce a novel but straightforward characterization of resolvability for hypercubes. We propose an integer linear programming method to assess resolvability rapidly, and provide a more costly but definite method based on Gröbner bases to determine whether or not a set of vertices resolves an arbitrary Hamming graph. As proof of concept, we identify a resolving set of size 77 in the metric space of all octapeptides (i.e., proteins composed of eight amino acids) with respect to the Hamming distance; in particular, any octamer may be readily represented as a 77-dimensional real-vector. Representing k-mers as low-dimensional numerical vectors may enable new applications of machine learning algorithms to symbolic sequences.

cs.DM↗
arXiv2019

Online Sparse Subspace Clustering

This paper focuses on the sparse subspace clustering problem, and develops an online algorithmic solution to cluster data points on-the-fly, without revisiting the whole dataset. The strategy involves an online solution of a sparse representation (SR) problem to build a (sparse) dictionary of similarities where points in the same subspace are considered "similar," followed by a spectral clustering based on the obtained similarity matrix. When the SR cost is strongly convex, the online solution converges to within a neighborhood of the optimal time-varying batch solution. A dynamic regret analysis is performed when the SR cost is not strongly convex.

math.OC↗
arXiv2019

Stochastic Subspace Descent

We present two stochastic descent algorithms that apply to unconstrained optimization and are particularly efficient when the objective function is slow to evaluate and gradients are not easily obtained, as in some PDE-constrained optimization and machine learning problems. The basic algorithm projects the gradient onto a random subspace at each iteration, similar to coordinate descent but without restricting directional derivatives to be along the axes. This algorithm is previously known but we provide new analysis. We also extend the popular SVRG method to this framework but without requiring that the objective function be written as a finite sum. We provide proofs of convergence for our methods under various convexity assumptions and show favorable results when compared to gradient descent and BFGS on non-convex problems from the machine learning and shape optimization literature. We also note that our analysis gives a proof that the iterates of SVRG and several other popular first-order stochastic methods, in their original formulation, converge almost surely to the optimum; to our knowledge, prior to this work the iterates of SVRG had only been known to converge in expectation.

math.OC↗
arXiv2019

Penalized basis models for very large spatial datasets

Many modern spatial models express the stochastic variation component as a basis expansion with random coefficients. Low rank models, approximate spectral decompositions, multiresolution representations, stochastic partial differential equations and empirical orthogonal functions all fall within this basic framework. Given a particular basis, stochastic dependence relies on flexible modeling of the coefficients. Under a Gaussianity assumption, we propose a graphical model family for the stochastic coefficients by parameterizing the precision matrix. Sparsity in the precision matrix is encouraged using a penalized likelihood framework. Computations follow from a majorization-minimization approach, a byproduct of which is a connection to the graphical lasso. The result is a flexible nonstationary spatial model that is adaptable to very large datasets. We apply the model to two large and heterogeneous spatial datasets in statistical climatology and recover physically sensible graphical structures. Moreover, the model performs competitively against the popular LatticeKrig model in predictive cross-validation, but substantially improves the Akaike information criterion score.

stat.ME↗