arXiv subjects

James Saunderson

Publications and source records attributed to James Saunderson.

At least 19 recordsLinked to original sources

arXiv2026

Non-negative polynomials without hyperbolic certificates of non-negativity

In this paper we study the relationship between the set of all non-negative multivariate homogeneous polynomials and those, which we call hyperwrons, whose non-negativity can be deduced from an identity involving the Wronskians of hyperbolic polynomials. We give a sufficient condition on positive integers $m$ and $2y$ such that there are non-negative polynomials of degree $2y$ in $m$ variables that are not hyperwrons. Furthermore, we give an explicit example of a non-negative quartic form that is not a sum of hyperwrons. We partially extend our results to hyperzouts, which are polynomials whose non-negativity can be deduced from an identity involving the Bézoutians of hyperbolic polynomials.

math.OC↗

arXiv2026

Correction of Pooling Matrix Mis-specifications in Compressed Sensing Based Group Testing

Compressed sensing, which involves the reconstruction of sparse signals from an under-determined linear system, has been recently used to solve problems in group testing. In a public health context, group testing aims to determine the health status values of p subjects from n<<p pooled tests, where a pool is defined as a mixture of small, equal-volume portions of the samples of a subset of subjects. This approach saves on the number of tests administered in pandemics or other resource-constrained scenarios. In practical group testing in time-constrained situations, a technician can inadvertently make a small number of errors during pool preparation, which leads to errors in the pooling matrix, which we term `model mismatch errors' (MMEs). This poses difficulties while determining health status values of the participating subjects from the results on n<<p pooled tests. In this paper, we present an algorithm to correct the MMEs in the pooled tests directly from the pooled results and the available (inaccurate) pooling matrix. Our approach then reconstructs the signal vector from the corrected pooling matrix, in order to determine the health status of the subjects. We further provide theoretical guarantees for the correction of the MMEs and the reconstruction error from the corrected pooling matrix. We also provide several supporting numerical results.

stat.AP↗

arXiv2025

Sparsification of sums with respect to convex cones

Let $x_1,x_2,\ldots,x_m$ be elements of a convex cone $K$ such that their sum, $e$, is in the relative interior of $K$. An $ε$-sparsification of the sum involves taking a subset of the $x_i$ and reweighting them by positive scalars, so that the resulting sum is $ε$-close to $e$, where error is measured in a relative sense with respect to the order induced by $K$. This generalizes the influential spectral sparsification model for sums of positive semidefinite matrices. This paper introduces and studies the sparsification function of a convex cone, which measures, in the worst case over all possible sums from the cone, the smallest size of an $ε$-sparsifier. The linear-sized spectral sparsification theorem of Batson, Spielman, and Srivastava can be viewed as a bound on the sparsification function of the cone of positive semidefinite matrices. This result is generalized to a family of convex cones (including all hyperbolicity cones) that admit a $ν$-logarithmically homogeneous self-concordant barrier with certain additional properties. For these cones, the sparsification function is bounded above by $\lceil4ν/ε^2\rceil$. For general convex cones that only admit an ordinary $ν$-logarithmically homogeneous self-concordant barrier, the sparsification function is bounded above by $\lceil(4ν/ε)^2\rceil$. Furthermore, the paper explores how sparsification functions interact with various convex geometric operations (such as conic lifts), and describes implications of sparsification with respect to cones for certain conic optimization problems.

math.OC↗

arXiv2025

Robust Non-adaptive Group Testing under Errors in Group Membership Specifications

Given $p$ samples, each of which may or may not be defective, group testing (GT) aims to determine their defect status by performing tests on $n < p$ `groups', where a group is formed by mixing a subset of the $p$ samples. Assuming that the number of defective samples is very small compared to $p$, GT algorithms have provided excellent recovery of the status of all $p$ samples with even a small number of groups. Most existing methods, however, assume that the group memberships are accurately specified. This assumption may not always be true in all applications, due to various resource constraints. Such errors could occur, eg, when a technician, preparing the groups in a laboratory, unknowingly mixes together an incorrect subset of samples as compared to what was specified. We develop a new GT method, the Debiased Robust Lasso Test Method (DRLT), that handles such group membership specification errors. The proposed DRLT method is based on an approach to debias, or reduce the inherent bias in, estimates produced by Lasso, a popular and effective sparse regression technique. We also provide theoretical upper bounds on the reconstruction error produced by our estimator. Our approach is then combined with two carefully designed hypothesis tests respectively for (i) the identification of defective samples in the presence of errors in group membership specifications, and (ii) the identification of groups with erroneous membership specifications. The DRLT approach extends the literature on bias mitigation of statistical estimators such as the LASSO, to handle the important case when some of the measurements contain outliers, due to factors such as group membership specification errors. We present numerical results which show that our approach outperforms several baselines and robust regression techniques for identification of defective samples as well as erroneously specified groups.

stat.ML↗

arXiv2025

QICS: Quantum Information Conic Solver

We introduce QICS (Quantum Information Conic Solver), an open-source primal-dual interior point solver fully implemented in Python, which is focused on solving optimization problems arising in quantum information theory. QICS has the ability to solve optimization problems involving the quantum relative entropy, noncommutative perspectives of operator convex functions, and related functions. It also includes an efficient semidefinite programming solver which exploits sparsity, as well as support for Hermitian matrices. QICS is also currently supported by the Python optimization modelling software PICOS. This paper aims to document the implementation details of the algorithm and cone oracles used in QICS, and serve as a reference guide for the software. Additionally, we showcase extensive numerical experiments which demonstrate that QICS outperforms state-of-the-art quantum relative entropy programming solvers, and has comparable performance to state-of-the-art semidefinite programming solvers.

math.OC↗

arXiv2025

A Feedback Control Framework for Incentivised Suburban Parking Utilisation and Urban Core Traffic Relief

Urban traffic congestion, exacerbated by inefficient parking management and cruising for parking, significantly hampers mobility and sustainability in smart cities. Drivers often face delays searching for parking spaces, influenced by factors such as accessibility, cost, distance, and available services such as charging facilities in the case of electric vehicles. These inefficiencies contribute to increased urban congestion, fuel consumption, and environmental impact. Addressing these challenges, this paper proposes a feedback control incentivisation-based system that aims to better distribute vehicles between city and suburban parking facilities offering park-and-charge/-ride services. Individual driver behaviours are captured via discrete choice models incorporating factors of importance to parking location choice among drivers, such as distance to work, public transport connectivity, charging infrastructure availability, and amount of incentive offered; and are regulated through principles of ergodic control theory. The proposed framework is applied to an electric vehicle park-and-charge/-ride problem, and demonstrates how predictable long-term behaviour of the system can be guaranteed.

eess.SY↗

arXiv2025

Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems

Quantum relative entropy optimization refers to a class of convex problems in which a linear functional is minimized over an affine section of the epigraph of the quantum relative entropy function. Recently, the self-concordance of a natural barrier function was proved for this set, and various implementations of interior-point methods have been made available to solve this class of optimization problems. In this paper, we show how common structures arising from applications in quantum information theory can be exploited to improve the efficiency of solving quantum relative entropy optimization problems using interior-point methods. First, we show that the natural barrier function for the epigraph of the quantum relative entropy composed with positive linear operators is self-concordant, even when these linear operators map to singular matrices. Compared to modelling problems using the full quantum relative entropy cone, this allows us to remove redundant log-determinant expressions from the barrier function and reduce the overall barrier parameter. Second, we show how certain slices of the quantum relative entropy cone exhibit useful properties which should be exploited whenever possible to perform certain key steps of interior-point methods more efficiently. We demonstrate how these methods can be applied to applications in quantum information theory, including quantifying quantum key rates, quantum rate-distortion functions, quantum channel capacities, and the ground state energy of Hamiltonians. Our numerical results show that these techniques improve computation times by up to several orders of magnitude, and allow previously intractable problems to be solved.

quant-ph↗

arXiv2025

Fast Debiasing of the LASSO Estimator

In high-dimensional sparse regression, the \textsc{Lasso} estimator offers excellent theoretical guarantees but is well-known to produce biased estimates. To address this, \cite{Javanmard2014} introduced a method to ``debias" the \textsc{Lasso} estimates for a random sub-Gaussian sensing matrix $\boldsymbol{A}$. Their approach relies on computing an ``approximate inverse" $\boldsymbol{M}$ of the matrix $\boldsymbol{A}^\top \boldsymbol{A}/n$ by solving a convex optimization problem. This matrix $\boldsymbol{M}$ plays a critical role in mitigating bias and allowing for construction of confidence intervals using the debiased \textsc{Lasso} estimates. However the computation of $\boldsymbol{M}$ is expensive in practice as it requires iterative optimization. In the presented work, we re-parameterize the optimization problem to compute a ``debiasing matrix" $\boldsymbol{W} := \boldsymbol{AM}^{\top}$ directly, rather than the approximate inverse $\boldsymbol{M}$. This reformulation retains the theoretical guarantees of the debiased \textsc{Lasso} estimates, as they depend on the \emph{product} $\boldsymbol{AM}^{\top}$ rather than on $\boldsymbol{M}$ alone. Notably, we provide a simple, computationally efficient, closed-form solution for $\boldsymbol{W}$ under similar conditions for the sensing matrix $\boldsymbol{A}$ used in the original debiasing formulation, with an additional condition that the elements of every row of $\boldsymbol{A}$ have uncorrelated entries. Also, the optimization problem based on $\boldsymbol{W}$ guarantees a unique optimal solution, unlike the original formulation based on $\boldsymbol{M}$. We verify our main result with numerical simulations.

stat.ML↗

arXiv2025

Operator convexity along lines, self-concordance, and sandwiched Rényi entropies

Barrier methods play a central role in the theory and practice of convex optimization. One of the most general and successful analyses of barrier methods for convex optimization, due to Nesterov and Nemirovskii, relies on the notion of self-concordance. While an extremely powerful concept, proving self-concordance of barrier functions can be very difficult. In this paper we give a simple way to verify that the natural logarithmic barrier of a convex nonlinear constraint is self-concordant via the theory of operator convex functions. Namely, we show that if a convex function is operator convex along any one-dimensional restriction, then the natural logarithmic barrier of its epigraph is self-concordant. We apply this technique to construct self-concordant barriers for the epigraphs of functions arising in quantum information theory. Notably, we apply this to the sandwiched Rényi entropy function, for which no self-concordant barrier was known before. Additionally, we utilize our sufficient condition to provide simplified proofs for previously established self-concordance results for the noncommutative perspective of operator convex functions. An implementation of the convex cones considered in this paper is now available in our open source interior-point solver QICS.

math.OC↗

arXiv2025

Inner approximations of convex sets and intersections of projectionally exposed cones

A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper we construct two p-exposed cones in dimension $5$ whose intersection is not p-exposed. This construction also leads to the first example of an amenable cone that is not projectionally exposed, showing that these properties, which coincide in dimension at most $4$, are distinct in dimension $5$. In order to achieve these goals, we develop a new technique for constructing arbitrarily tight inner convex approximations of compact convex sets with desired facial structure. These inner approximations have the property that all proper faces are extreme points, with the exception of a specific exposed face of the original set.

math.OC↗

arXiv2024

A Bregman Proximal Perspective on Classical and Quantum Blahut-Arimoto Algorithms

The Blahut-Arimoto algorithm is a well-known method to compute classical channel capacities and rate-distortion functions. Recent works have extended this algorithm to compute various quantum analogs of these quantities. In this paper, we show how these Blahut-Arimoto algorithms are special instances of mirror descent, which is a type of Bregman proximal method, and a well-studied generalization of gradient descent for constrained convex optimization. Using recently developed convex analysis tools, we show how analysis based on relative smoothness and strong convexity recovers known sublinear and linear convergence rates for Blahut-Arimoto algorithms. This Bregman proximal viewpoint allows us to derive related algorithms with similar convergence guarantees to solve problems in information theory for which Blahut-Arimoto-type algorithms are not directly applicable. We apply this framework to compute energy-constrained classical and quantum channel capacities, classical and quantum rate-distortion functions, and approximations of the relative entropy of entanglement, all with provable convergence guarantees.

cs.IT↗

arXiv2024

On noisy duplication channels with Markov sources

Channels with noisy duplications have recently been used to model the nanopore sequencer. This paper extends some foundational information-theoretic results to this new scenario. We prove the asymptotic equipartition property (AEP) for noisy duplication processes based on ergodic Markov processes. A consequence is that the noisy duplication channel is information stable for ergodic Markov sources, and therefore the channel capacity constrained to Markov sources is the Markov-constrained Shannon capacity. We use the AEP to estimate lower bounds on the capacity of the binary symmetric channel with Bernoulli and geometric duplications using Monte Carlo simulations. In addition, we relate the AEP for noisy duplication processes to the AEP for hidden semi-Markov processes.

cs.IT↗

arXiv2024

Efficient Computation of the Quantum Rate-Distortion Function

The quantum rate-distortion function plays a fundamental role in quantum information theory, however there is currently no practical algorithm which can efficiently compute this function to high accuracy for moderate channel dimensions. In this paper, we show how symmetry reduction can significantly simplify common instances of the entanglement-assisted quantum rate-distortion problems. This allows us to better understand the properties of the quantum channels which obtain the optimal rate-distortion trade-off, while also allowing for more efficient computation of the quantum rate-distortion function regardless of the numerical algorithm being used. Additionally, we propose an inexact variant of the mirror descent algorithm to compute the quantum rate-distortion function with provable sublinear convergence rates. We show how this mirror descent algorithm is related to Blahut-Arimoto and expectation-maximization methods previously used to solve similar problems in information theory. Using these techniques, we present the first numerical experiments to compute a multi-qubit quantum rate-distortion function, and show that our proposed algorithm solves faster and to higher accuracy when compared to existing methods.

quant-ph↗

arXiv2023

Compressive quantum waveform estimation

Quantum waveform estimation, in which quantum sensors sample entire time series, promises to revolutionize the sensing of weak and stochastic signals, such as the biomagnetic impulses emitted by firing neurons. For long duration signals with rapid transients, regular quantum sampling becomes prohibitively resource intensive as it demands many measurements with distinct control and readout. In this Manuscript, we demonstrate how careful choice of quantum measurements, along with the modern mathematics of compressive sensing, achieves quantum waveform estimation of sparse signals in a number of measurements far below the Nyquist requirement. We sense synthesized neural-like magnetic signals with radiofrequency-dressed ultracold atoms, retrieving successful waveform estimates with as few measurements as compressive theoretical bounds guarantee.

quant-ph↗

arXiv2023

An Inexact Frank-Wolfe Algorithm for Composite Convex Optimization Involving a Self-Concordant Function

In this paper, we consider Frank-Wolfe-based algorithms for composite convex optimization problems with objective involving a logarithmically-homogeneous, self-concordant functions. Recent Frank-Wolfe-based methods for this class of problems assume an oracle that returns exact solutions of a linearized subproblem. We relax this assumption and propose a variant of the Frank-Wolfe method with inexact oracle for this class of problems. We show that our inexact variant enjoys similar convergence guarantees to the exact case, while allowing considerably more flexibility in approximately solving the linearized subproblem. In particular, our approach can be applied if the subproblem can be solved prespecified additive error or to prespecified relative error (even though the optimal value of the subproblem may not be uniformly bounded). Furthermore, our approach can also handle the situation where the subproblem is solved via a randomized algorithm that fails with positive probability. Our inexact oracle model is motivated by certain large-scale semidefinite programs where the subproblem reduces to computing an extreme eigenvalue-eigenvector pair, and we demonstrate the practical performance of our algorithm with numerical experiments on problems of this form.

math.OC↗

arXiv2023

Hyperbolicity cones are amenable

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or "nice") which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.

math.OC↗

arXiv2023

Optimal self-concordant barriers for quantum relative entropies

Quantum relative entropies are jointly convex functions of two positive definite matrices that generalize the Kullback-Leibler divergence and arise naturally in quantum information theory. In this paper, we prove self-concordance of natural barrier functions for the epigraphs of various quantum relative entropies and divergences. Furthermore we show that these barriers have optimal barrier parameter. These barriers allow convex optimization problems involving quantum relative entropies to be directly solved using interior point methods for non-symmetric cones, avoiding the approximations and lifting techniques used in previous approaches. More generally, we establish the self-concordance of natural barriers for various closed convex cones related to the noncommutative perspectives of operator concave functions, and show that the resulting barrier parameters are optimal.

math.OC↗

arXiv2022

Finite-State Semi-Markov Channels for Nanopore Sequencing

Nanopore sequencing is an emerging DNA sequencing technology that has been proposed for use in DNA storage systems. We propose the noisy nanopore channel model for nanopore sequencing. This model captures duplications, inter-symbol interference, and noisy measurements by concatenating an i.i.d. duplication channel with a finite-state semi-Markov channel. Compared to previous models, this channel models the dominant distortions of the nanopore while remaining tractable. Anticipating future coding schemes, we derive MAP detection algorithms and estimate achievable rates. Given that finite-state semi-Markov channels are a subclass of channels with memory, we conjecture that the achievable rate of the noisy nanopore channel can be optimised using a variation of the generalised Blahut-Arimoto algorithm.

cs.IT↗