arXiv subjects

Mark A. Davenport

Publications and source records attributed to Mark A. Davenport.

At least 37 records · Page 2Linked to original sources

arXiv2020

Semi-supervised sequence classification through change point detection

Sequential sensor data is generated in a wide variety of practical applications. A fundamental challenge involves learning effective classifiers for such sequential data. While deep learning has led to impressive performance gains in recent years in domains such as speech, this has relied on the availability of large datasets of sequences with high-quality labels. In many applications, however, the associated class labels are often extremely limited, with precise labelling/segmentation being too expensive to perform at a high volume. However, large amounts of unlabeled data may still be available. In this paper we propose a novel framework for semi-supervised learning in such contexts. In an unsupervised manner, change point detection methods can be used to identify points within a sequence corresponding to likely class changes. We show that change points provide examples of similar/dissimilar pairs of sequences which, when coupled with labeled, can be used in a semi-supervised classification setting. Leveraging the change points and labeled data, we form examples of similar/dissimilar sequences to train a neural network to learn improved representations for classification. We provide extensive synthetic simulations and show that the learned representations are superior to those learned through an autoencoder and obtain improved results on both simulated and real-world human activity recognition datasets.

cs.LG↗

arXiv2020

Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in $\ell_2(\mathbb{Z})$ which are strictly bandlimited to a frequency band $[-W,W]$ and maximally concentrated in a time interval $\{0,\ldots,N-1\}$. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in $\mathbb{C}^N$ whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band $[-W,W]$. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior -- slightly fewer than $2NW$ eigenvalues are very close to $1$, slightly fewer than $N-2NW$ eigenvalues are very close to $0$, and very few eigenvalues are not near $1$ or $0$. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near $0$ or $1$. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between $ε$ and $1-ε$. Also, we obtain bounds detailing how close the first $\approx 2NW$ eigenvalues are to $1$ and how close the last $\approx N-2NW$ eigenvalues are to $0$. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between $ε$ and $1-ε$.

math.CA↗

arXiv2020

Simultaneous Preference and Metric Learning from Paired Comparisons

A popular model of preference in the context of recommendation systems is the so-called \emph{ideal point} model. In this model, a user is represented as a vector $\mathbf{u}$ together with a collection of items $\mathbf{x_1}, \ldots, \mathbf{x_N}$ in a common low-dimensional space. The vector $\mathbf{u}$ represents the user's "ideal point," or the ideal combination of features that represents a hypothesized most preferred item. The underlying assumption in this model is that a smaller distance between $\mathbf{u}$ and an item $\mathbf{x_j}$ indicates a stronger preference for $\mathbf{x_j}$. In the vast majority of the existing work on learning ideal point models, the underlying distance has been assumed to be Euclidean. However, this eliminates any possibility of interactions between features and a user's underlying preferences. In this paper, we consider the problem of learning an ideal point representation of a user's preferences when the distance metric is an unknown Mahalanobis metric. Specifically, we present a novel approach to estimate the user's ideal point $\mathbf{u}$ and the Mahalanobis metric from paired comparisons of the form "item $\mathbf{x_i}$ is preferred to item $\mathbf{x_j}$." This can be viewed as a special case of a more general metric learning problem where the location of some points are unknown a priori. We conduct extensive experiments on synthetic and real-world datasets to exhibit the effectiveness of our algorithm.

stat.ML↗

arXiv2020

Dynamic Knowledge embedding and tracing

The goal of knowledge tracing is to track the state of a student's knowledge as it evolves over time. This plays a fundamental role in understanding the learning process and is a key task in the development of an intelligent tutoring system. In this paper we propose a novel approach to knowledge tracing that combines techniques from matrix factorization with recent progress in recurrent neural networks (RNNs) to effectively track the state of a student's knowledge. The proposed \emph{DynEmb} framework enables the tracking of student knowledge even without the concept/skill tag information that other knowledge tracing models require while simultaneously achieving superior performance. We provide experimental evaluations demonstrating that DynEmb achieves improved performance compared to baselines and illustrating the robustness and effectiveness of the proposed framework. We also evaluate our approach using several real-world datasets showing that the proposed model outperforms the previous state-of-the-art. These results suggest that combining embedding models with sequential models such as RNNs is a promising new direction for knowledge tracing.

cs.IR↗

arXiv2020

Low-rank matrix completion and denoising under Poisson noise

This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

stat.ML↗

arXiv2019

Sparse Bayesian Learning with Dynamic Filtering for Inference of Time-Varying Sparse Signals

Many signal processing applications require estimation of time-varying sparse signals, potentially with the knowledge of an imperfect dynamics model. In this paper, we propose an algorithm for dynamic filtering of time-varying sparse signals based on the sparse Bayesian learning (SBL) framework. The key idea underlying the algorithm, termed SBL-DF, is the incorporation of a signal prediction generated from a dynamics model and estimates of previous time steps into the hyperpriors of the SBL probability model. The proposed algorithm is robust to imperfect dynamics models (due to the propagation of dynamics information through higher-order statistics), robust to certain undesirable dictionary properties such as coherence (due to properties of the SBL framework), allows the use of arbitrary dynamics models, and requires the tuning of fewer parameters than many other dynamic filtering algorithms do. We also extend the fast marginal likelihood SBL inference procedure to the informative hyperprior setting to create a particularly efficient version of the SBL-DF algorithm. Numerical simulations show that SBL-DF converges much faster and to more accurate solutions than standard SBL and other dynamical filtering algorithms. In particular, we show that SBL-DF outperforms state of the art algorithms when the dictionary contains challenging structure such as coherence and column scaling, as is found in many practical applications.

eess.SP↗

arXiv2019

Active embedding search via noisy paired comparisons

Suppose that we wish to estimate a user's preference vector $w$ from paired comparisons of the form "does user $w$ prefer item $p$ or item $q$?," where both the user and items are embedded in a low-dimensional Euclidean space with distances that reflect user and item similarities. Such observations arise in numerous settings, including psychometrics and psychology experiments, search tasks, advertising, and recommender systems. In such tasks, queries can be extremely costly and subject to varying levels of response noise; thus, we aim to actively choose pairs that are most informative given the results of previous comparisons. We provide new theoretical insights into the benefits and challenges of greedy information maximization in this setting, and develop two novel strategies that maximize lower bounds on information gain and are simpler to analyze and compute respectively. We use simulated responses from a real-world dataset to validate our strategies through their similar performance to greedy information maximization, and their superior preference estimation over state-of-the-art selection methods as well as random queries.

stat.ML↗

arXiv2019

Trading beams for bandwidth: Imaging with randomized beamforming

We study the problem of actively imaging a range-limited far-field scene using an antenna array. We describe how the range limit imposes structure in the measurements across multiple wavelengths. This structure allows us to introduce a novel trade-off: the number of spatial array measurements (i.e., beams that have to be formed) can be reduced significantly lower than the number array elements if the scene is illuminated with a broadband source. To take advantage of this trade-off, we use a small number of "generic" linear combinations of the array outputs, instead of the phase offsets used in conventional beamforming. We provide theoretical justification for the proposed trade-off without making any strong structural assumptions on the target scene (such as sparsity) except that it is range limited. In proving our theoretical results, we take inspiration from the sketching literature. We also provide simulation results to establish the merit of the proposed signal acquisition strategy. Our proposed method results in a reduction in the number of required spatial measurements in an array imaging system and hence can directly impact their speed and cost of operation.

eess.SP↗

arXiv2018

Estimation of Poisson arrival processes under linear models

In this paper we consider the problem of estimating the parameters of a Poisson arrival process where the rate function is assumed to lie in the span of a known basis. Our goal is to estimate the basis expansions coefficients given a realization of this process. We establish novel guarantees concerning the accuracy achieved by the maximum likelihood estimate. Our initial result is near-optimal, with the exception of an undesirable dependence on the dynamic range of the rate function. We then show how to remove this dependence through a process of "noise regularization", which results in an improved bound. We conjecture that a similar guarantee should be possible when using a more direct (deterministic) regularization scheme. We conclude with a discussion of practical applications and an empirical examination of the proposed regularization schemes.

cs.IT↗

arXiv2018

ROAST: Rapid Orthogonal Approximate Slepian Transform

In this paper, we provide a Rapid Orthogonal Approximate Slepian Transform (ROAST) for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal. The ROAST offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence (DPSS) vectors (also known as Slepian basis vectors). As such, the ROAST is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading DPSS vectors themselves. Moreover, the subspace angle between the ROAST subspace and the corresponding DPSS subspace can be made arbitrarily small. The complexity of computing the representation of a signal using the ROAST is comparable to the FFT, which is much less than the complexity of using the DPSS basis vectors. We also give non-asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors, but also that it can provide high-quality approximations of all sampled sinusoids within the band of interest.

cs.IT↗

arXiv2018

On the stability and accuracy of least squares approximations

We consider the problem of reconstructing an unknown function $f$ on a domain $X$ from samples of $f$ at $n$ randomly chosen points with respect to a given measure $ρ_X$. Given a sequence of linear spaces $(V_m)_{m>0}$ with ${\rm dim}(V_m)=m\leq n$, we study the least squares approximations from the spaces $V_m$. It is well known that such approximations can be inaccurate when $m$ is too close to $n$, even when the samples are noiseless. Our main result provides a criterion on $m$ that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in $L^2(X,ρ_X)$, is comparable to the best approximation error of $f$ by elements from $V_m$. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with $ρ_X$ being the uniform measure, and algebraic polynomials, with $ρ_X$ being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.

math.NA↗

arXiv2017

The Fast Slepian Transform

The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). We show that there exist fast constructions for computing approximate projections onto the leading Slepian basis elements. The complexity of the resulting algorithms is comparable to the FFT, and scales favorably as the quality of the desired approximation is increased. In the process of bounding the complexity of these algorithms, we also establish new nonasymptotic results on the eigenvalue distribution of discrete time-frequency localization operators. We then demonstrate how these algorithms allow us to efficiently compute the solution to certain least-squares problems that arise in signal processing. We also provide simulations comparing these fast, approximate Slepian methods to exact Slepian methods as well as the traditional FFT based methods.

math.NA↗

arXiv2017

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

Bandlimiting and timelimiting operators play a fundamental role in analyzing bandlimited signals that are approximately timelimited (or vice versa). In this paper, we consider a time-frequency (in the discrete Fourier transform (DFT) domain) limiting operator whose eigenvectors are known as the periodic discrete prolate spheroidal sequences (PDPSSs). We establish new nonasymptotic results on the eigenvalue distribution of this operator. As a byproduct, we also characterize the eigenvalue distribution of a set of submatrices of the DFT matrix, which is of independent interest.

cs.IT↗

arXiv2016

Dynamic matrix recovery from incomplete observations under an exact low-rank constraint

Low-rank matrix factorizations arise in a wide variety of applications -- including recommendation systems, topic models, and source separation, to name just a few. In these and many other applications, it has been widely noted that by incorporating temporal information and allowing for the possibility of time-varying models, significant improvements are possible in practice. However, despite the reported superior empirical performance of these dynamic models over their static counterparts, there is limited theoretical justification for introducing these more complex models. In this paper we aim to address this gap by studying the problem of recovering a dynamically evolving low-rank matrix from incomplete observations. First, we propose the locally weighted matrix smoothing (LOWEMS) framework as one possible approach to dynamic matrix recovery. We then establish error bounds for LOWEMS in both the {\em matrix sensing} and {\em matrix completion} observation models. Our results quantify the potential benefits of exploiting dynamic constraints both in terms of recovery accuracy and sample complexity. To illustrate these benefits we provide both synthetic and real-world experimental results.

stat.ML↗

arXiv2016

Constrained adaptive sensing

Suppose that we wish to estimate a vector $\mathbf{x} \in \mathbb{C}^n$ from a small number of noisy linear measurements of the form $\mathbf{y} = \mathbf{A x} + \mathbf{z}$, where $\mathbf{z}$ represents measurement noise. When the vector $\mathbf{x}$ is sparse, meaning that it has only $s$ nonzeros with $s \ll n$, one can obtain a significantly more accurate estimate of $\mathbf{x}$ by adaptively selecting the rows of $\mathbf{A}$ based on the previous measurements provided that the signal-to-noise ratio (SNR) is sufficiently large. In this paper we consider the case where we wish to realize the potential of adaptivity but where the rows of $\mathbf{A}$ are subject to physical constraints. In particular, we examine the case where the rows of $\mathbf{A}$ are constrained to belong to a finite set of allowable measurement vectors. We demonstrate both the limitations and advantages of adaptive sensing in this constrained setting. We prove that for certain measurement ensembles, the benefits offered by adaptive designs fall far short of the improvements that are possible in the unconstrained adaptive setting. On the other hand, we also provide both theoretical and empirical evidence that in some scenarios adaptivity does still result in substantial improvements even in the constrained setting. To illustrate these potential gains, we propose practical algorithms for constrained adaptive sensing by exploiting connections to the theory of optimal experimental design and show that these algorithms exhibit promising performance in some representative applications.

cs.IT↗

arXiv2016

An overview of low-rank matrix recovery from incomplete observations

Low-rank matrices play a fundamental role in modeling and computational methods for signal processing and machine learning. In many applications where low-rank matrices arise, these matrices cannot be fully sampled or directly observed, and one encounters the problem of recovering the matrix given only incomplete and indirect observations. This paper provides an overview of modern techniques for exploiting low-rank structure to perform matrix recovery in these settings, providing a survey of recent advances in this rapidly-developing field. Specific attention is paid to the algorithms most commonly used in practice, the existing theoretical guarantees for these algorithms, and representative practical applications of these techniques.

cs.IT↗

arXiv2014

1-Bit Matrix Completion

In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question we ask is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but we show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when ||M||_{\infty} <= α, and rank(M) <= r. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. In addition, we also show that if instead of recovering M we simply wish to obtain an estimate of the distribution generating the 1-bit measurements, then we can eliminate the requirement that ||M||_{\infty} <= α. For both cases, we provide lower bounds showing that these estimates are near-optimal. We conclude with a suite of experiments that both verify the implications of our theorems as well as illustrate some of the practical applications of 1-bit matrix completion. In particular, we compare our program to standard matrix completion methods on movie rating data in which users submit ratings from 1 to 5. In order to use our program, we quantize this data to a single bit, but we allow the standard matrix completion program to have access to the original ratings (from 1 to 5). Surprisingly, the approach based on binary data performs significantly better.

math.ST↗

arXiv2013

Signal Space CoSaMP for Sparse Recovery with Redundant Dictionaries

Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well-conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary. In this paper we describe a variant of the iterative recovery algorithm CoSaMP for this more challenging setting. We utilize the D-RIP, a condition on the sensing matrix analogous to the well-known restricted isometry property. In contrast to prior work, the method and analysis are "signal-focused"; that is, they are oriented around recovering the signal rather than its dictionary coefficients. Under the assumption that we have a near-optimal scheme for projecting vectors in signal space onto the model family of candidate sparse signals, we provide provable recovery guarantees. Developing a practical algorithm that can provably compute the required near-optimal projections remains a significant open problem, but we include simulation results using various heuristics that empirically exhibit superior performance to traditional recovery algorithms.

cs.IT↗