arXiv subjects

Manuel Fernandez

Publications and source records attributed to Manuel Fernandez.

6 recordsLinked to original sources

arXiv2024

The smallest singular value of inhomogenous random rectangular matrices

Let $A \in \mathbb{R}^{N \times n}$ ($N \geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\beta$ moment. We show that the smallest singular value $\sigma_n(A)$ satisfies \[ \Pr \left(\sigma_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon)^{N-n+1} + e^{-cN}, \] for all $\varepsilon > 0$, where $c,C$ depend only on $\beta$ and the $2+\beta$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. When the $2+\beta$ moment assumption is replaced with a uniform anti-concentration assumption, $\sup_z \Pr\left(|X-z| < a\right) < b$, we show that \[ \Pr\left(\sigma_n(A) \leq \varepsilon(\sqrt{N+1} - \sqrt{n})\right) \leq (C\varepsilon\log(1/\varepsilon))^{N-n+1} + e^{-cN}, \] where $c,C$ now depend only on $a$ and $b$. This extends more recent work of Livshyts, whose showed that such lower tail estimates held for rectrangular matrices with i.i.d. rows. To prove these results we employ a number of new technical ingredients, including a new deviation inequality for the regularized Hilbert-Schmidt norm and a recently proven small ball estimate for the distance between a random vector and a subspace spanned by an inhomogeneous rectangular matrix.

math.PR↗

arXiv2024

The Black Hole Explorer: Back End Electronics

This paper describes specification and early design of back end signal processing subsystems for the Black Hole Explorer (BHEX) Very Long Baseline Interferometry (VLBI) space telescope. The "back end" consists of two subsystems. First, the block downconverter (BDC) is a heterodyne system that performs a frequency translation of the analog signal from IF to baseband and amplifies and filters it for digitization. Second, the digital back end (DBE) samples the analog signal with an analog-to-digital converters (ADC) and digitally processes the data stream formatting them to the VLBI "VDIF" standard and converting to Ethernet packets for 100 gigabit-per-second (Gb/s) Ethernet transport to the optical downlink system. Both the BDC and the DBE for BHEX support eight channels of 4.096 GHz bandwidth each, for a total processed bandwidth of 32.768 GHz. The BHEX back end benefits from mature terrestrial back end heritage, described in some detail. The BHEX back end itself is in the early stages of design, with requirements, interface specifications, and component trade studies well advanced. The aim is to build a prototype using terrestrial grade parts which are available in functionally identical space grade equivalents, and to use this prototype to advance the back end Technology Readiness Level (TRL) preparing for a Small Explorer (SMEX) proposal in 2025.

astro-ph.IM↗

arXiv2019

Graph Spanners in the Message-Passing Model

Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been extensively studied. We study the problem of computing a graph spanner when the edges of the input graph are distributed across two or more sites in an arbitrary, possibly worst-case partition, and the goal is for the sites to minimize the communication used to output a spanner. We assume the message-passing model of communication, for which there is a point-to-point link between all pairs of sites as well as a coordinator who is responsible for producing the output. We stress that the subset of edges that each site has is not related to the network topology, which is fixed to be point-to-point. While this model has been extensively studied for related problems such as graph connectivity, it has not been systematically studied for graph spanners. We present the first tradeoffs for total communication versus the quality of the spanners computed, for two or more sites, as well as for additive and multiplicative notions of distortion. We show separations in the communication complexity when edges are allowed to occur on multiple sites, versus when each edge occurs on at most one site. We obtain nearly tight bounds (up to polylog factors) for the communication of additive $2$-spanners in both the with and without duplication models, multiplicative $(2k-1)$-spanners in the with duplication model, and multiplicative $3$ and $5$-spanners in the without duplication model. Our lower bound for multiplicative $3$-spanners employs biregular bipartite graphs rather than the usual Erdős girth conjecture graphs and may be of wider interest.

cs.DS↗

arXiv2019

The Query Complexity of Mastermind with $\ell_p$ Distances

Consider a variant of the Mastermind game in which queries are $\ell_p$ distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector $\mathbf{y}\in\{-k,-k+1,\dots,k-1,k\}^n$ and answers to queries of the form $\Vert\mathbf{y}-\mathbf{x}\Vert_p$ where $\mathbf{x}\in\{-k,-k+1,\dots,k-1,k\}^n$. The goal is to minimize the number of queries made in order to correctly guess $\mathbf{y}$. Motivated by this question, in this work, we develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e.\ coordinate-wise sums of functions of the absolute value. This in particular includes distances such as the smooth max (LogSumExp) as well as many widely-studied $M$-estimator losses, such as $\ell_p$ norms, the $\ell_1$-$\ell_2$ loss, the Huber loss, and the Fair estimator loss. When we apply this result to $\ell_p$ queries, we obtain an upper bound of $O\left(\min\left\{n,\frac{n\log k}{\log n}\right\}\right)$ queries for any real $1\leq p<\infty$. We also show matching lower bounds up to constant factors for the $\ell_p$ problem, even for adaptive algorithms for the approximation version of the problem, in which the problem is to output $\mathbf{y}'$ such that $\Vert\mathbf{y}'-\mathbf{y}\Vert_p\leq R$ for any $R\leq k^{1-\varepsilon}n^{1/p}$ for constant $\varepsilon>0$. Thus, essentially any approximation of this problem is as hard as finding the hidden vector exactly, up to constant factors. Finally, we show that for the noisy version of the problem, i.e. the setting when the codemaker answers queries with any $q = (1\pm\varepsilon)\Vert\mathbf{y}-\mathbf{x}\Vert_p$, there is no query efficient algorithm.

cs.DS↗

arXiv2019

Tight Kernel Query Complexity of Kernel Ridge Regression and Kernel $k$-means Clustering

We present tight lower bounds on the number of kernel evaluations required to approximately solve kernel ridge regression (KRR) and kernel $k$-means clustering (KKMC) on $n$ input points. For KRR, our bound for relative error approximation to the minimizer of the objective function is $Ω(nd_{\mathrm{eff}}^λ/\varepsilon)$ where $d_{\mathrm{eff}}^λ$ is the effective statistical dimension, which is tight up to a $\log(d_{\mathrm{eff}}^λ/\varepsilon)$ factor. For KKMC, our bound for finding a $k$-clustering achieving a relative error approximation of the objective function is $Ω(nk/\varepsilon)$, which is tight up to a $\log(k/\varepsilon)$ factor. Our KRR result resolves a variant of an open question of El Alaoui and Mahoney, asking whether the effective statistical dimension is a lower bound on the sampling complexity or not. Furthermore, for the important practical case when the input is a mixture of Gaussians, we provide a KKMC algorithm which bypasses the above lower bound.

cs.DS↗

arXiv2004

The Gauss-Landau-Hall problem on Riemannian surfaces

We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.

math-ph↗