arXiv subjects

Mehrdad Ghadiri

Publications and source records attributed to Mehrdad Ghadiri.

At least 19 recordsLinked to original sources

arXiv2026

Weibull-Stationary Stochastic Differential Equations for Conditional Long-Horizon Wind Power Forecasting

We present a one-month-ahead conditional probabilistic framework for wind-power forecasting at ten-minute resolution. Monthly Weibull shape and scale parameters are estimated from serially dependent SCADA wind-speed data, corrected through a Godambe covariance, and forecast by a heteroskedastic Kalman filter on a bivariate VAR(1) state-space model. Conditional on the MMSE forecasted Weibull invariant law, we construct and compare three positive wind-speed SDE models: an Ornstein-Uhlenbeck-Weibull transform, a Fokker-Planck drift-first diffusion, and a Fokker-Planck diffusion-first model. The simulated wind-speed ensembles are mapped to power through a calibrated XGBoost power curve. Applied to January 2021 data from a Senvion MM92 turbine at Kelmarsh Wind Farm, the three SDE formulations are statistically indistinguishable in probabilistic accuracy, with mean CRPS values between 1.569 and 1.575 m/s. The diffusion-first model is therefore preferred on computational grounds, reducing runtime by about a factor of seven relative to the OU-Weibull model. In the power domain, the Wasserstein distance between simulated and observed distributions is 26.1-27.6 kW, below $1.4\%$ of rated capacity, while the monthly energy-yield bias is about $-7.3\%$ for the examined month. Exceedance-probability errors remain below 1.6 percentage points over the 0-1500 kW range and about 2.2 percentage points near rated power. These quantities provide decision-relevant probabilistic inputs for downstream operational problems, rather than completed reserve, storage, market, or fatigue-optimization decisions. Full marginalisation over the Kalman predictive law of the Weibull parameters is left as a natural extension.

stat.AP↗

arXiv2026

Unbiased Regression-Adjusted Estimation of Average Treatment Effects in Randomized Controlled Trials

This article introduces a leave-one-out regression adjustment (LOORA) for estimating average treatment effects in randomized controlled trials. In finite samples, LOORA removes the bias of conventional regression adjustment and yields exact variance formulas for regression-adjusted Horvitz-Thompson and difference-in-means estimators. Ridge regularization curbs the influence of high-leverage observations, improving stability and precision in small samples. In large samples, LOORA matches the variance of the regression-adjusted estimator in Lin (2013) while remaining exactly unbiased. Two within-subject experimental applications, each providing a realistic joint distribution of potential outcomes as ground truth, show that LOORA removes substantial bias and achieves confidence interval coverage close to the nominal level.

econ.EM↗

arXiv2025

Entrywise Approximate Solutions for SDDM Systems in Almost-Linear Time

We present an algorithm that given any invertible symmetric diagonally dominant M-matrix (SDDM), i.e., a principal submatrix of a graph Laplacian, $\boldsymbol{\mathit{L}}$ and a nonnegative vector $\boldsymbol{\mathit{b}}$, computes an entrywise approximation to the solution of $\boldsymbol{\mathit{L}} \boldsymbol{\mathit{x}} = \boldsymbol{\mathit{b}}$ in $\tilde{O}(m n^{o(1)})$ time with high probability, where $m$ is the number of nonzero entries and $n$ is the dimension of the system.

cs.DS↗

arXiv2025

Entrywise Approximation for Matrix Inversion and Linear Systems

We study the bit complexity of inverting diagonally dominant matrices, which are associated with random walk quantities such as hitting times and escape probabilities. Such quantities can be exponentially small, even on undirected unit-weighted graphs. However, their nonnegativity suggests that they can be approximated entrywise, leading to a stronger notion of approximation than vector norm-based error. Under this notion of error, existing Laplacian solvers and fast matrix multiplication approaches have bit complexities of $mn^2$ and $n^{ω+1}$, respectively, where $m$ is the number of nonzero entries in the matrix, $n$ is its size, and $ω$ is the matrix multiplication exponent. We present algorithms that compute entrywise $\exp(ε)$-approximate inverses of row diagonally dominant $L$-matrices (RDDL) in two settings: (1) when the matrix entries are given in floating-point representation; (2) when they are given in fixed-point representation. For floating-point inputs, we present a cubic-time algorithm and show that it has an optimal running time under the all-pairs shortest paths (APSP) conjecture. For fixed-point inputs, we present several algorithms for solving linear systems and inverting RDDL and SDDM matrices, all with high probability. Omitting logarithmic factors: (1) For SDDM matrices, we provide an algorithm for solving a linear system with entrywise approximation guarantees using $\tilde{O}(m\sqrt{n})$ bit operations, and another for computing an entrywise approximate inverse using $\tilde{O}(mn)$ bit operations. (2) For RDDL matrices, we present an algorithm for solving a linear system using $\tilde{O}(mn^{1+o(1)})$ bit operations, and two algorithms for computing an entrywise approximate inverse: one using $\tilde{O}(n^{ω+0.5})$ bit operations, and the other using $\tilde{O}(mn^{1.5+o(1)})$ bit operations.

cs.DS↗

arXiv2025

Fast Tensor Completion via Approximate Richardson Iteration

We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods to solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is often lost in TC regression problems, making direct extensions unclear. This work proposes a novel lifting method for approximately solving TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We analyze the convergence rate of our approximate Richardson iteration-based algorithm, and our empirical study shows that it can be 100x faster than direct methods for CP completion on real-world tensors.

cs.DS↗

arXiv2025

A Tight Lower Bound for the Approximation Guarantee of Higher-Order Singular Value Decomposition

We prove that the classic approximation guarantee for the higher-order singular value decomposition (HOSVD) is tight by constructing a tensor for which HOSVD achieves an approximation ratio of $N/(1+\varepsilon)$, for any $\varepsilon > 0$. This matches the upper bound of De Lathauwer et al. (2000a) and shows that the approximation ratio of HOSVD cannot be improved. Using a more advanced construction, we also prove that the approximation guarantees for the ST-HOSVD algorithm of Vannieuwenhoven et al. (2012) and higher-order orthogonal iteration (HOOI) of De Lathauwer et al. (2000b) are tight by showing that they can achieve their worst-case approximation ratio of $N / (1 + \varepsilon)$, for any $\varepsilon > 0$.

cs.DS↗

arXiv2025

A Note on the Stability of the Sherman-Morrison-Woodbury Formula

We study the numerical stability of the Sherman-Morrison-Woodbury (SMW) identity. Let $B = A + UV^T$ and assume $U$ and $V$ both have full-column rank. We explore error bounds for the SMW identity when we are only able to compute approximate inverses. For both forward and backward errors, we present upper bounds as a function of the two-norm error of the approximate inverses. We verify with numerical experiments that, in certain cases, our bounds accurately capture the behavior of the errors.

math.NA↗

arXiv2024

Entrywise Approximate Laplacian Solving

We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require $n$ bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.

cs.DS↗

arXiv2024

Improving the Bit Complexity of Communication for Distributed Convex Optimization

We consider the communication complexity of some fundamental convex optimization problems in the point-to-point (coordinator) and blackboard communication models. We strengthen known bounds for approximately solving linear regression, $p$-norm regression (for $1\leq p\leq 2$), linear programming, minimizing the sum of finitely many convex nonsmooth functions with varying supports, and low rank approximation; for a number of these fundamental problems our bounds are nearly optimal, as proven by our lower bounds. Among our techniques, we use the notion of block leverage scores, which have been relatively unexplored in this context, as well as dropping all but the ``middle" bits in Richardson-style algorithms. We also introduce a new communication problem for accurately approximating inner products and establish a lower bound using the spherical Radon transform. Our lower bound can be used to show the first separation of linear programming and linear systems in the distributed model when the number of constraints is polynomial, addressing an open question in prior work.

cs.DS↗

arXiv2024

The Bit Complexity of Dynamic Algebraic Formulas and their Determinants

Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are only analyzed under exact arithmetic (real-RAM model and finite fields) which may not accurately reflect the complexity guarantees of real computers. In this paper, we analyze the stability and bit complexity of such data structures for expressions that involve the inversion, multiplication, addition, and subtraction of matrices under the word-RAM model. We show that the bit complexity only increases linearly in the number of matrix operations in the expression. In addition, we consider the bit complexity of maintaining the determinant of a matrix expression. We show that the required bit complexity depends on the logarithm of the condition number of matrices instead of the logarithm of their determinant. We also discuss rank maintenance and its connections to determinant maintenance. Our results have wide applications ranging from computational geometry (e.g., computing the volume of a polytope) to optimization (e.g., solving linear programs using the simplex algorithm).

cs.CC↗

arXiv2023

On Symmetric Factorizations of Hankel Matrices

We present two conjectures regarding the running time of computing symmetric factorizations for a Hankel matrix $\mathbf{H}$ and its inverse $\mathbf{H}^{-1}$ as $\mathbf{B}\mathbf{B}^*$ under fixed-point arithmetic. If solved, these would result in a faster-than-matrix-multiplication algorithm for solving sparse poly-conditioned linear programming problems, a fundamental problem in optimization and theoretical computer science. To justify our proposed conjectures and running times, we show weaker results of computing decompositions of the form $\mathbf{B}\mathbf{B}^* - \mathbf{C}\mathbf{C}^*$ for Hankel matrices and their inverses with the same running time. In addition, to promote our conjectures further, we discuss the connections of Hankel matrices and their symmetric factorizations to sum-of-squares (SoS) decompositions of single-variable polynomials.

math.NA↗

arXiv2023

Subquadratic Kronecker Regression with Applications to Tensor Decomposition

Kronecker regression is a highly-structured least squares problem $\min_{\mathbf{x}} \lVert \mathbf{K}\mathbf{x} - \mathbf{b} \rVert_{2}^2$, where the design matrix $\mathbf{K} = \mathbf{A}^{(1)} \otimes \cdots \otimes \mathbf{A}^{(N)}$ is a Kronecker product of factor matrices. This regression problem arises in each step of the widely-used alternating least squares (ALS) algorithm for computing the Tucker decomposition of a tensor. We present the first subquadratic-time algorithm for solving Kronecker regression to a $(1+\varepsilon)$-approximation that avoids the exponential term $O(\varepsilon^{-N})$ in the running time. Our techniques combine leverage score sampling and iterative methods. By extending our approach to block-design matrices where one block is a Kronecker product, we also achieve subquadratic-time algorithms for (1) Kronecker ridge regression and (2) updating the factor matrices of a Tucker decomposition in ALS, which is not a pure Kronecker regression problem, thereby improving the running time of all steps of Tucker ALS. We demonstrate the speed and accuracy of this Kronecker regression algorithm on synthetic data and real-world image tensors.

cs.DS↗

arXiv2023

The Bit Complexity of Efficient Continuous Optimization

We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number of arithmetic operations, they match the current time complexity of multiplying two $n$-by-$n$ matrices (up to polylogarithmic factors). However, previous work has typically assumed infinite precision arithmetic, and due to complicated inverse maintenance techniques, the actual running times of these algorithms are unknown. To settle the running time and bit complexity of these algorithms, we demonstrate that a core common subroutine, known as \emph{inverse maintenance}, is backward-stable. Additionally, we show that iterative approaches for solving constrained weighted regression problems can be accomplished with bounded-error pre-conditioners. Specifically, we prove that linear programs can be solved approximately in matrix multiplication time multiplied by polylog factors that depend on the condition number $κ$ of the matrix and the inner and outer radius of the LP problem. $p$-norm regression can be solved approximately in matrix multiplication time multiplied by polylog factors in $κ$. Lastly, linear regression can be solved approximately in input-sparsity time multiplied by polylog factors in $κ$. Furthermore, we present results for achieving lower than matrix multiplication time for $p$-norm regression by utilizing faster solvers for sparse linear systems.

cs.DS↗

arXiv2023

Approximately Optimal Core Shapes for Tensor Decompositions

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

cs.DS↗

arXiv2022

Constant-Factor Approximation Algorithms for Socially Fair $k$-Clustering

We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups, whose special cases include the socially fair $k$-median ($p=1$) and socially fair $k$-means ($p=2$) problems. We present (1) a polynomial-time $(5+2\sqrt{6})^p$-approximation with at most $k+m$ centers (2) a $(5+2\sqrt{6}+ε)^p$-approximation with $k$ centers in time $n^{2^{O(p)}\cdot m^2}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{m}\cdot\text{poly}(n)$. The first result is obtained via a refinement of the iterative rounding method using a sequence of linear programs. The latter two results are obtained by converting a solution with up to $k+m$ centers to one with $k$ centers using sparsification methods for (2) and via an exhaustive search for (3). We also compare the performance of our algorithms with existing bicriteria algorithms as well as exactly $k$ center approximation algorithms on benchmark datasets, and find that our algorithms also outperform existing methods in practice.

cs.DS↗

arXiv2022

Amortized Rejection Sampling in Universal Probabilistic Programming

Naive approaches to amortized inference in probabilistic programs with unbounded loops can produce estimators with infinite variance. This is particularly true of importance sampling inference in programs that explicitly include rejection sampling as part of the user-programmed generative procedure. In this paper we develop a new and efficient amortized importance sampling estimator. We prove finite variance of our estimator and empirically demonstrate our method's correctness and efficiency compared to existing alternatives on generative programs containing rejection sampling loops and discuss how to implement our method in a generic probabilistic programming framework.

cs.LG↗

arXiv2021

Faster $p$-Norm Regression Using Sparsity

For a matrix $A\in \mathbb{R}^{n\times d}$ with $n\geq d$, we consider the dual problems of $\min \|Ax-b\|_p^p, \, b\in \mathbb{R}^n$ and $\min_{A^\top x=b} \|x\|_p^p,\, b\in \mathbb{R}^d$. We improve the runtimes for solving these problems to high accuracy for every $p>1$ for sufficiently sparse matrices. We show that recent progress on fast sparse linear solvers can be leveraged to obtain faster than matrix-multiplication algorithms for any $p > 1$, i.e., in time $\tilde{O}(pn^θ)$ for some $θ< ω$, the matrix multiplication constant. We give the first high-accuracy input sparsity $p$-norm regression algorithm for solving $\min \|Ax-b\|_p^p$ with $1 < p \leq 2$, via a new row sampling theorem for the smoothed $p$-norm function. This algorithm runs in time $\tilde{O}(\text{nnz}(A) + d^4)$ for any $1<p\leq 2$, and in time $\tilde{O}(\text{nnz}(A) + d^θ)$ for $p$ close to $2$, improving on the previous best bound where the exponent of $d$ grows with $\max\{p, p/(p-1)\}$.

cs.DS↗

arXiv2021

Fast Low-Rank Tensor Decomposition by Ridge Leverage Score Sampling

Low-rank tensor decomposition generalizes low-rank matrix approximation and is a powerful technique for discovering low-dimensional structure in high-dimensional data. In this paper, we study Tucker decompositions and use tools from randomized numerical linear algebra called ridge leverage scores to accelerate the core tensor update step in the widely-used alternating least squares (ALS) algorithm. Updating the core tensor, a severe bottleneck in ALS, is a highly-structured ridge regression problem where the design matrix is a Kronecker product of the factor matrices. We show how to use approximate ridge leverage scores to construct a sketched instance for any ridge regression problem such that the solution vector for the sketched problem is a $(1+\varepsilon)$-approximation to the original instance. Moreover, we show that classical leverage scores suffice as an approximation, which then allows us to exploit the Kronecker structure and update the core tensor in time that depends predominantly on the rank and the sketching parameters (i.e., sublinear in the size of the input tensor). We also give upper bounds for ridge leverage scores as rows are removed from the design matrix (e.g., if the tensor has missing entries), and we demonstrate the effectiveness of our approximate ridge regressioni algorithm for large, low-rank Tucker decompositions on both synthetic and real-world data.

cs.DS↗