arXiv subjects

Rikhav Shah

Publications and source records attributed to Rikhav Shah.

At least 19 recordsLinked to original sources

arXiv2026

Entry growth in Gaussian elimination

Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.

math.NA↗

arXiv2026

What is Jackson's constant?

We prove a refinement of Jackson's theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson's construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson's construction in this respect.

math.NA↗

arXiv2026

Spectral density estimation for normal matrices

The spectral density estimation problem asks for an algorithm that, given an $n\times n$ matrix $A$, outputs a probability measure that is a good approximation to the uniform distribution on the eigenvalues of $A$, called the spectral density of $A$. This paper considers the setting where $A$ is a large normal matrix that is accessible only through matrix-vector product queries. We provide an algorithm that makes just $m$ matrix-vector queries to $A$ and returns, with high probability, a measure within earth mover's distance $O(1/m+\log m/{\sqrt n})$ of the true spectral density of $A$. We provide a complementary lower bound that any algorithm producing an $\varepsilon$-approximation to the true spectral density for large matrices must make $\Omega(1/\varepsilon)$ matrix-vector queries. The lower bound holds even for the more restricted case of real symmetric input matrices. In combination with our upper bound, it shows that spectral density estimation is essentially no harder for complex normal matrices than for real symmetric matrices.

math.NA↗

arXiv2026

DynMuon: A Dynamic Spectral Shaping View of Muon

In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix $M=U\Sigma V^\top$ with its polar factor $UV^\top$. In this work, we consider a class of Muon-like updates, where we replace the update $M$ with $U\Sigma^p V^\top$ for some parameter $p$. We call this a "spectral-shaping" operation, and develop a theory of how to pick $p$ which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive $p$ helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative $p$ helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules $p$ from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss. Our code is available at https://github.com/fzwark/DynMuon.

cs.LG↗

arXiv2026

The maximum-entropy median-martingale

This short note explores the maximum-entropy walk on the unit interval that is a median-martingale. That is, the median of its next state is equal to its current state. The stationary distribution of this walk is the arcsine distribution, and we provide a proof that elucidates the connection to two classical arcsine laws for Brownian motion. The notion of a martingale is further generalized, and a larger class of walks is considered and similarly characterized.

math.PR↗

arXiv2026

On the exponential rate of the condition number of Fourier submatrices and Vandermonde matrices

The discrete Fourier transform matrix is one of the most important matrices in linear algebra, and submatrices of it arise in a variety of applications. Though the discrete Fourier transform matrix is unitary, its submatrices can be exponentially ill-conditioned, an obstacle to accurate computation. This work resolves the exact rate of the exponential ill-conditioning for square submatrices with contiguous rows and columns. As a consequence, we obtain a tight upper bound of $2 G/\pi$ on the exponential rate for all submatrices with contiguous columns, or, equivalently, all Vandermonde submatrices with distinct support points, where $G$ is Catalan's constant. These results follow from a more general analysis of Vandermonde and Vandermonde-like matrices for which exact estimates for exponential ill-conditioning are developed in terms of logarithmic potentials.

math.NA↗

arXiv2026

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions. (Updated to reflect the status of the open problems as of August 20, 2026.)

math.NA↗

arXiv2025

High-Performance Imaging in a Dilution Refrigerator

Nanophotonic light-matter interfaces hold great promise for quantum technologies. Enhancing local electromagnetic fields, they enable highly efficient detectors, can help realize optically connected processors, or serve as quantum repeaters. In-situ fiber-coupling at sub-Kelvin temperatures, as required for test and development of new devices, proves challenging as suitable cryogenic microscopes are not readily available. Here, we report on a robust and versatile confocal imaging system integrated in a dilution refrigerator, enabling high-resolution visualization of nanophotonic structures on a transparent diamond substrate. Our imaging system achieves a resolution of 1.1 {\mu}m and a field-of-view of 2.5 mm. The system requires no movable parts at cryogenic temperatures and features a large working distance, thereby allowing optical and microwave probe access, as well as direct anchoring of temperature sensitive samples to a cold finger, needed for applications with high thermal load. This system will facilitate the development of scalable, integrated quantum optics technology, as required for research on large scale quantum networks.

quant-ph↗

arXiv2025

Even Faster Kernel Matrix Linear Algebra via Density Estimation

This paper studies the use of kernel density estimation (KDE) for linear algebraic tasks involving the kernel matrix of a collection of $n$ data points in $\mathbb R^d$. In particular, we improve upon existing algorithms for computing the following up to $(1+\varepsilon)$ relative error: matrix-vector products, matrix-matrix products, the spectral norm, and sum of all entries. The runtimes of our algorithms depend on the dimension $d$, the number of points $n$, and the target error $\varepsilon$. Importantly, the dependence on $n$ in each case is far lower when accessing the kernel matrix through KDE queries as opposed to reading individual entries. Our improvements over existing best algorithms (particularly those of Backurs, Indyk, Musco, and Wagner '21) for these tasks reduce the polynomial dependence on $\varepsilon$, and additionally decreases the dependence on $n$ in the case of computing the sum of all entries of the kernel matrix. We complement our upper bounds with several lower bounds for related problems, which provide (conditional) quadratic time hardness results and additionally hint at the limits of KDE based approaches for the problems we study.

cs.DS↗

arXiv2025

Simulation and measurement of Black Body Radiation background in a Transition Edge Sensor

The Any Light Particle Search II (ALPS II) experiment at DESY, Hamburg, is a Light-Shining-through-a-Wall (LSW) experiment aiming to probe the existence of axions and axion-like particles (ALPs), which are candidates for dark matter. Data collection in ALPS II is underway utilizing a heterodyne-based detection scheme. A complementary run for confirmation or as an alternative method is planned using single photon detection, requiring a sensor capable of measuring low-energy photons ($1064\,\mathrm{nm}$, $1.165\,\mathrm{eV}$) with high efficiency (higher than $50\,\%$) and a low background rate (below $7.7\cdot10^{-6}\,\mathrm{cps}$). To meet these requirements, we are investigating a tungsten Transition Edge Sensor (TES) provided by NIST, which operates in its superconducting transition region at millikelvin temperatures. This sensor exploits the drastic change in resistance caused by the absorption of a single photon. We find that the background observed in the setup with a fiber-coupled TES is consistent with Black Body Radiation (BBR) as the primary background contributor. A framework was developed to simulate BBR propagation to the TES under realistic conditions. The framework not only allows the exploration of background reduction strategies, such as improving the TES energy resolution, but also reproduces, within uncertainties, the spectral distribution of the observed background. These simulations have been validated with experimental data, in agreement with the modeled background distribution, and show that the improved energy resolution reduces the background rate in the $1064\,\mathrm{nm}$ signal region by one order of magnitude, to approximately $10^{-4}\,\mathrm{cps}$. However, this rate must be reduced further to meet the ALPS II requirements.

hep-ex↗

arXiv2025

Matrix Factorizations with Uniformly Random Pivoting

This paper highlights a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which admits a unified analysis of the entire class of algorithms. The result is the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. One important consequence of this randomized pivoting rule is a provable polynomial bound on the numerical stability of the Jacobi eigenvalue algorithm without any preconditioning, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

math.NA↗

arXiv2025

The Pseudospectrum of Random Compressions of Matrices

The compression of a matrix $A\in\mathbb C^{n\times n}$ onto a subspace $V\subset\mathbb C^n$ is the matrix $Q^*AQ$ where the columns of $Q$ form an orthonormal basis for $V$. This is an important object in both operator theory and numerical linear algebra. Of particular interest are the eigenvalues of the compression and their stability under perturbations. This paper considers compressions onto subspaces sampled from the Haar measure on the complex Grassmannian. We show the expected area of the $\varepsilon$-pseudospectrum of such compressions is bounded by $\text{poly}(n)\log^2(1/\varepsilon)\cdot\varepsilon^\beta$, where $\beta=6/5,4/3$, or $2$ depending on some mild assumptions on $A$. Along the way, we obtain (a) tail bounds for the least singular value of compressions and (b) non-asymptotic small-ball estimates for random non-Hermitian quadratic forms surpassing bounds achieved by existing methods.

math.PR↗

arXiv2024

Sparse Pseudospectral Shattering

The eigenvalues and eigenvectors of nonnormal matrices can be unstable under perturbations of their entries. This renders an obstacle to the analysis of numerical algorithms for non-Hermitian eigenvalue problems. A recent technique to handle this issue is pseudospectral shattering [BGVKS23], showing that adding a random perturbation to any matrix has a regularizing effect on the stability of the eigenvalues and eigenvectors. Prior work has analyzed the regularizing effect of dense Gaussian perturbations, where independent noise is added to every entry of a given matrix [BVKS20, BGVKS23, BKMS21, JSS21]. We show that the same effect can be achieved by adding a sparse random perturbation. In particular, we show that given any $n\times n$ matrix $M$ of polynomially bounded norm: (a) perturbing $O(n\log^2(n))$ random entries of $M$ by adding i.i.d. complex Gaussians yields $\log\kappa_V(A)=O(\text{poly}\log(n))$ and $\log (1/\eta(A))=O(\text{poly}\log(n))$ with high probability; (b) perturbing $O(n^{1+\alpha})$ random entries of $M$ for any constant $\alpha>0$ yields $\log\kappa_V(A)=O_\alpha(\log(n))$ and $\log(1/\eta(A))=O_\alpha(\log(n))$ with high probability. Here, $\kappa_V(A)$ denotes the condition number of the eigenvectors of the perturbed matrix $A$ and $\eta(A)$ denotes its minimum eigenvalue gap. A key mechanism of the proof is to reduce the study of $\kappa_V(A)$ to control of the pseudospectral area and minimum eigenvalue gap of $A$, which are further reduced to estimates on the least two singular values of shifts of $A$. We obtain the required least singular value estimates via a streamlining of an argument of Tao and Vu [TV07] specialized to the case of sparse complex Gaussian perturbations. [Rest of abstract in pdf].

math.PR↗

arXiv2024

A Kaczmarz-Inspired Method for Orthogonalization

This paper asks if the following iterative procedure approximately orthogonalizes a set of $n$ linearly independent unit vectors while preserving their span: in each iteration, access a random pair of vectors and replace one with the component perpendicular to the other, renormalized to be a unit vector. We provide a positive answer: any given set of starting vectors converges almost surely to an orthonormal basis of their span. We specifically argue that the $n$-volume of the parallelepiped generated by the vectors approaches 1 (i.e. the parallelepiped approaches a hypercube). If $A$ is the matrix formed by taking these vectors as columns, this volume is simply $\det(|A|)$ where $|A|=(A^*A)^{1/2}$. We show that $O(n^2\log(1/(\det(|A|)\varepsilon)))$ iterations suffice to bring ${\det(|A|)}$ above $1-\varepsilon$ with constant probability.

math.PR↗

arXiv2024

Fast Hermitian Diagonalization with Nearly Optimal Precision

Algorithms for numerical tasks in finite precision simultaneously seek to minimize the number of floating point operations performed, and also the number of bits of precision required by each floating point operation. This paper presents an algorithm for Hermitian diagonalization requiring only $\lg(1/\varepsilon)+O(\log(n)+\log\log(1/\varepsilon))$ bits of precision where $n$ is the size of the input matrix and $\varepsilon$ is the target error. Furthermore, it runs in near matrix multiplication time. In the general setting, the first complete analysis of the stability of a near matrix multiplication time algorithm for diagonalization is that of Banks et al. [BGVKS20]. They exhibit an algorithm for diagonalizing an arbitrary matrix up to $\varepsilon$ backward error using only $O(\log^4(n/\varepsilon)\log(n))$ bits of precision. This work focuses on the Hermitian setting, where we determine a dramatically improved bound on the number of bits needed. In particular, the result is close to providing a practical bound. The exact bit count depends on the specific implementation of matrix multiplication and QR decomposition one wishes to use, but if one uses suitable $O(n^3)$-time implementations, then for $\varepsilon=10^{-15},n=4000$, we show 92 bits of precision suffice (and 59 are necessary). By comparison, the same parameters in [BGVKS20] does not even show that 682,916,525,000 bits suffice.

math.NA↗

arXiv2024

Achieving the Highest Possible Elo Rating

Elo rating systems measure the approximate skill of each competitor in a game or sport. A competitor's rating increases when they win and decreases when they lose. Increasing one's rating can be difficult work; one must hone their skills and consistently beat the competition. Alternatively, with enough money you can rig the outcome of games to boost your rating. This paper poses a natural question for Elo rating systems: say you manage to get together $n$ people (including yourself) and acquire enough money to rig $k$ games. How high can you get your rating, asymptotically in $k$? In this setting, the people you gathered aren't very interested in the game, and will only play if you pay them to. This paper resolves the question for $n=2$ up to constant additive error, and provide close upper and lower bounds for all other $n$, including for $n$ growing arbitrarily with $k$. There is a phase transition at $n=k^{1/3}$: there is a huge increase in the highest possible Elo rating from $n=2$ to $n=k^{1/3}$, but (depending on the particular Elo system used) little-to-no increase for any higher $n$. Past the transition point $n>k^{1/3}$, the highest possible Elo is at least $Θ(k^{1/3})$. The corresponding upper bound depends on the particular system used, but for the standard Elo system, is $Θ(k^{1/3}\log(k)^{1/3})$.

math.CO↗

arXiv2023

A first application of machine and deep learning for background rejection in the ALPS II TES detector

Axions and axion-like particles are hypothetical particles predicted in extensions of the standard model and are promising cold dark matter candidates. The Any Light Particle Search (ALPS II) experiment is a light-shining-through-the-wall experiment that aims to produce these particles from a strong light source and magnetic field and subsequently detect them through a reconversion into photons. With an expected rate $\sim$ 1 photon per day, a sensitive detection scheme needs to be employed and characterized. One foreseen detector is based on a transition edge sensor (TES). Here, we investigate machine and deep learning algorithms for the rejection of background events recorded with the TES. We also present a first application of convolutional neural networks to classify time series data measured with the TES.

hep-ex↗

arXiv2022

A Spectral Approach to Polytope Diameter

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.

math.CO↗