Related papers: Faster Dynamic Matrix Inverse for Faster LPs
In this paper, we consider the following inverse maintenance problem: given $A \in \mathbb{R}^{n\times d}$ and a number of rounds $r$, we receive a $n\times n$ diagonal matrix $D^{(k)}$ at round $k$ and we wish to maintain an efficient…
The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet well-understood,…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested…
We present a simple formula to update the pseudoinverse of a full-rank rectangular matrix that undergoes a low-rank modification, and demonstrate its utility for solving least squares problems. The resulting algorithm can be dramatically…
Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear…
In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain…
This paper introduces a new robust interior point method analysis for semidefinite programming (SDP). This new robust analysis can be combined with either logarithmic barrier or hybrid barrier. Under this new framework, we can improve the…
Inverse linear programming (LP) has received increasing attention due to its potential to generate efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and…
We present a general framework that utilizes different efficient data structures to improve various sparsification problems involving an iterative process. We also provide insights and characterization for different iterative process, and…
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…
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…
We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same…
This paper shows how to solve linear programs of the form $\min_{Ax=b,x\geq0} c^\top x$ with $n$ variables in time $$O^*((n^{\omega}+n^{2.5-\alpha/2}+n^{2+1/6}) \log(n/\delta))$$ where $\omega$ is the exponent of matrix multiplication,…
In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive…
Many convex problems in machine learning and computer science share the same form: \begin{align*} \min_{x} \sum_{i} f_i( A_i x + b_i), \end{align*} where $f_i$ are convex functions on $\mathbb{R}^{n_i}$ with constant $n_i$, $A_i \in…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…
A fast metasurface optimization strategy for finite-size metasurfaces modeled using integral equations is presented. The metasurfaces considered are constructed from finite patterned metallic claddings supported by grounded dielectric…
We build upon the recent papers by Weinstein and Yu (FOCS'16), Larsen (FOCS'12), and Clifford et al. (FOCS'15) to present a general framework that gives amortized lower bounds on the update and query times of dynamic data structures. Using…
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…