arXiv subjects

Mark Davenport

Publications and source records attributed to Mark Davenport.

8 recordsLinked to original sources

arXiv2026

A Fast Broadband Beamspace Transformation

We present a new computationally efficient method for multi-beamforming in the broadband setting. Our "fast beamspace transformation" forms $B$ beams from $M$ sensor outputs using a number of operations per sample that scales linearly (to within logarithmic factors) with $M$ when $B\sim M$. While the narrowband version of this transformation can be performed efficiently with a spatial fast Fourier transform, the broadband setting requires coherent processing of multiple array snapshots simultaneously. Our algorithm works by taking $N$ samples off of each of $M$ sensors and encoding the sensor outputs into a set of coefficients using a special non-uniform spaced Fourier transform. From these coefficients, each beam is formed by solving a small system of equations that has Toeplitz structure. The total runtime complexity is $\mathcal{O}(M\log N+B\log N)$ operations per sample, exhibiting essentially the same scaling as in the narrowband case and vastly outperforming broadband beamformers based on delay and sum whose computations scale as $\mathcal{O}(MB)$. Alongside a careful mathematical formulation and analysis of our fast broadband beamspace transform, we provide a host of numerical experiments demonstrating the algorithm's favorable computational scaling and high accuracy. Finally, we demonstrate how tasks such as interpolating to ``off-grid" angles and nulling an interferer are more computationally efficient when performed directly in beamspace.

eess.SP↗

arXiv2023

Distance preservation in state-space methods for detecting causal interactions in dynamical systems

We analyze the popular ``state-space'' class of algorithms for detecting casual interaction in coupled dynamical systems. These algorithms are often justified by Takens' embedding theorem, which provides conditions under which relationships involving attractors and their delay embeddings are continuous. In practice, however, state-space methods often do not directly test continuity, but rather the stronger property of how these relationships preserve inter-point distances. This paper theoretically and empirically explores state-space algorithms explicitly from the perspective of distance preservation. We first derive basic theoretical guarantees applicable to simple coupled systems, providing conditions under which the distance preservation of a certain map reveals underlying causal structure. Second, we demonstrate empirically that typical coupled systems do not satisfy distance preservation assumptions. Taken together, our results underline the dependence of state-space algorithms on intrinsic system properties and the relationship between the system and the function used to measure it -- properties that are not directly associated with causal interaction.

eess.SP↗

arXiv2022

Iterative Broadband Source Localization

In this paper we consider the problem of localizing a set of broadband sources from a finite window of measurements. In the case of narrowband sources this can be reduced to the problem of spectral line estimation, where our goal is simply to estimate the active frequencies from a weighted mixture of pure sinusoids. There exists a plethora of modern and classical methods that effectively solve this problem. However, for a wide variety of applications the underlying sources are not narrowband and can have an appreciable amount of bandwidth. In this work, we extend classical greedy algorithms for sparse recovery (e.g., orthogonal matching pursuit) to localize broadband sources. We leverage models for samples of broadband signals based on a union of Slepian subspaces, which are more aptly suited for dealing with spectral leakage and dynamic range disparities. We show that by using these models, our adapted algorithms can successfully localize broadband sources under a variety of adverse operating scenarios. Furthermore, we show that our algorithms outperform complementary methods that use more standard Fourier models. We also show that we can perform estimation from compressed measurements with little loss in fidelity as long as the number of measurements are on the order of the signal's implicit degrees of freedom. We conclude with an in-depth application of these ideas to the problem of localization in multi-sensor arrays.

eess.SP↗

arXiv2022

Broadband Beamforming via Linear Embedding

In modern applications multi-sensor arrays are subject to an ever-present demand to accommodate signals with higher bandwidths. Standard methods for broadband beamforming, namely digital beamforming and true-time delay, are difficult and expensive to implement at scale. In this work, we explore an alternative method of broadband beamforming that uses a set of linear measurements and a robust low-dimensional signal subspace model. The linear measurements, taken directly from the sensors, serve as a method for dimensionality reduction and serve to limit the array readout. From these embedded samples, we show how the original samples can be recovered to within a provably small residual error using a Slepian subspace model. Previous work in multi-sensor array subspace models have largely analyzed performance from a qualitative or asymptotic perspective. In contrast, we give quantitative estimates of how well different dimensionality reduction strategies preserve the array gain. We also show how spatial and temporal correlations can be used to relax the standard Nyquist sampling criterion, how recovery can be achieved through fast algorithms, and how "hardware friendly" linear measurements can be designed.

eess.SP↗

arXiv2020

Generative causal explanations of black-box classifiers

We develop a method for generating causal post-hoc explanations of black-box classifiers based on a learned low-dimensional representation of the data. The explanation is causal in the sense that changing learned latent factors produces a change in the classifier output statistics. To construct these explanations, we design a learning framework that leverages a generative model and information-theoretic measures of causal influence. Our objective function encourages both the generative model to faithfully represent the data distribution and the latent factors to have a large causal influence on the classifier output. Our method learns both global and local explanations, is compatible with any classifier that admits class probabilities and a gradient, and does not require labeled attributes or knowledge of causal structure. Using carefully controlled test cases, we provide intuition that illuminates the function of our objective. We then demonstrate the practical utility of our method on image recognition tasks.

cs.LG↗

arXiv2020

Sample complexity and effective dimension for regression on manifolds

We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various implicit and explicit dimensionality-reduction methods that exploit manifold structure. Our first key contribution is to establish a novel nonasymptotic version of the Weyl law from differential geometry. From this we are able to show that certain spaces of smooth functions on a manifold are effectively finite-dimensional, with a complexity that scales according to the manifold dimension rather than any ambient data dimension. Finally, we show that given (potentially noisy) function values taken uniformly at random over a manifold, a kernel regression estimator (derived from the spectral decomposition of the manifold) yields minimax-optimal error bounds that are controlled by the effective dimension.

stat.ML↗

arXiv2018

A unified framework for manifold landmarking

The success of semi-supervised manifold learning is highly dependent on the quality of the labeled samples. Active manifold learning aims to select and label representative landmarks on a manifold from a given set of samples to improve semi-supervised manifold learning. In this paper, we propose a novel active manifold learning method based on a unified framework of manifold landmarking. In particular, our method combines geometric manifold landmarking methods with algebraic ones. We achieve this by using the Gershgorin circle theorem to construct an upper bound on the learning error that depends on the landmarks and the manifold's alignment matrix in a way that captures both the geometric and algebraic criteria. We then attempt to select landmarks so as to minimize this bound by iteratively deleting the Gershgorin circles corresponding to the selected landmarks. We also analyze the complexity, scalability, and robustness of our method through simulations, and demonstrate its superiority compared to existing methods. Experiments in regression and classification further verify that our method performs better than its competitors.

cs.LG↗

arXiv2012

On the Fundamental Limits of Adaptive Sensing

Suppose we can sequentially acquire arbitrary linear measurements of an n-dimensional vector x resulting in the linear model y = Ax + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy which cleverly selects the next row of A based on what has been previously observed should do far better than a nonadaptive strategy which sets the rows of A ahead of time, thus not trying to learn anything about the signal in between observations. This paper shows that the folk theorem is false. We prove that the advantages offered by clever adaptive strategies and sophisticated estimation procedures---no matter how intractable---over classical compressed acquisition/recovery schemes are, in general, minimal.

math.ST↗