arXiv subjects

Akshay Ramachandran

Publications and source records attributed to Akshay Ramachandran.

7 recordsLinked to original sources

arXiv2025

Optimal Bounds for Tyler's M-Estimator for Elliptical Distributions

A fundamental problem in statistics is estimating the shape matrix of an Elliptical distribution. This generalizes the familiar problem of Gaussian covariance estimation, for which the sample covariance achieves optimal estimation error. For Elliptical distributions, Tyler proposed a natural M-estimator and showed strong statistical properties in the asymptotic regime, independent of the underlying distribution. Numerical experiments show that this estimator performs very well, and that Tyler's iterative procedure converges quickly to the estimator. Franks and Moitra recently provided the first distribution-free error bounds in the finite sample setting, as well as the first rigorous convergence analysis of Tyler's iterative procedure. However, their results exceed the sample complexity of the Gaussian setting by a $\log^{2} d$ factor. We close this gap by proving optimal sample threshold and error bounds for Tyler's M-estimator for all Elliptical distributions, fully matching the Gaussian result. Moreover, we recover the algorithmic convergence even at this lower sample threshold. Our approach builds on the operator scaling connection of Franks and Moitra by introducing a novel pseudorandom condition, which we call $\infty$-expansion. We show that Elliptical distributions satisfy $\infty$-expansion at the optimal sample threshold, and then prove a novel scaling result for inputs satisfying this condition.

math.ST↗

arXiv2025

Computational aspects of the trace norm contraction coefficient

We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard. This contrasts with the classical analogue of this problem that can clearly be solved efficiently. We also establish the NP-hardness of deciding if the contraction coefficient is equal to 1, i.e., the channel can perfectly preserve a bit. As a consequence, deciding if a non-commutative graph has an independence number of at least 2 is NP-hard. In addition, we establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.

quant-ph↗

arXiv2024

Hidden convexity, optimization, and algorithms on rotation matrices

This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show that certain linear images of $\text{SO}(n)$ are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of $\text{SO}(n)$ is convex and that the projection of $\text{SO}(n)$ onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over $\text{SO}(n)$ with a single constraint or with constraints defined by low-rank matrices. Both of these results are optimal in a formal sense.

math.OC↗

arXiv2024

Strongly Polynomial Frame Scaling to High Precision

The frame scaling problem is: given vectors $U := \{u_{1}, ..., u_{n} \} \subseteq \mathbb{R}^{d}$, marginals $c \in \mathbb{R}^{n}_{++}$, and precision $\varepsilon > 0$, find left and right scalings $L \in \mathbb{R}^{d \times d}, r \in \mathbb{R}^n$ such that $(v_1,\dots,v_n) := (Lu_1 r_1,\dots,Lu_nr_n)$ simultaneously satisfies $\sum_{i=1}^n v_i v_i^{\mathsf{T}} = I_d$ and $\|v_{j}\|_{2}^{2} = c_{j}, \forall j \in [n]$, up to error $\varepsilon$. This problem has appeared in a variety of fields throughout linear algebra and computer science. In this work, we give a strongly polynomial algorithm for frame scaling with $\log(1/\varepsilon)$ convergence. This answers a question of Diakonikolas, Tzamos and Kane (STOC 2023), who gave the first strongly polynomial randomized algorithm with poly$(1/\varepsilon)$ convergence for the special case $c = \frac{d}{n} 1_{n}$. Our algorithm is deterministic, applies for general $c \in \mathbb{R}^{n}_{++}$, and requires $O(n^{3} \log(n/\varepsilon))$ iterations as compared to $O(n^{5} d^{11}/\varepsilon^{5})$ iterations of DTK. By lifting the framework of Linial, Samorodnitsky and Wigderson (Combinatorica 2000) for matrix scaling to frames, we are able to simplify both the algorithm and analysis. Our main technical contribution is to generalize the potential analysis of LSW to the frame setting and compute an update step in strongly polynomial time that achieves geometric progress in each iteration. In fact, we can adapt our results to give an improved analysis of strongly polynomial matrix scaling, reducing the $O(n^{5} \log(n/\varepsilon))$ iteration bound of LSW to $O(n^{3} \log(n/\varepsilon))$. Additionally, we prove a novel bound on the size of approximate frame scaling solutions, involving the condition measure $\barχ$ studied in the linear programming literature, which may be of independent interest.

cs.DS↗

arXiv2021

Near optimal sample complexity for matrix and tensor normal models via geodesic convexity

The matrix normal model, i.e., the family of Gaussian matrix-variate distributions whose covariance matrices are the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor normal models. For the above models, we show that the maximum likelihood estimator (MLE) achieves nearly optimal nonasymptotic sample complexity and nearly tight error rates in the Fisher-Rao and Thompson metrics. In contrast to prior work, our results do not rely on the factors being well-conditioned or sparse, nor do we need to assume an accurate enough initial guess. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bounds for the largest factor and for overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that the flip-flop algorithm, a practical and widely used iterative procedure to compute the MLE, converges linearly with high probability. Our main technical insight is that, given enough samples, the negative log-likelihood function is strongly geodesically convex in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels.

math.ST↗

arXiv2019

Spectral analysis of matrix scaling and operator scaling

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These bounds can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp-Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds.

cs.DS↗

arXiv2017

The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $ε$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \ldots, v_n \in \mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\inf_{v} \sum_{i=1}^n \| u_i - v_i \|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $ε$-nearly solution is at most $O({\rm{poly}}(d,n,ε))$ and there are $ε$-nearly solutions with squared distance at least $Ω(dε)$. The fundamental open question is whether the squared distance can be independent of the number of vectors $n$. We answer this question affirmatively by proving that the squared distance of any $ε$-nearly solution is $O(d^{13/2} ε)$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any $ε$-nearly solution is $O(d^2 n ε)$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $ε$-nearly solution is $O(d^{5/2} ε)$ when $n$ is large enough and $ε$ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.

cs.DS↗