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Support vector machine (SVM) is an optimal margin based classification technique in machine learning. SVM is a binary linear classifier which has been extended to non-linear data using Kernels and multi-class data using various techniques like one-versus-one, one-versus-rest, Crammer Singer SVM, Weston Watkins SVM and directed acyclic graph SVM (DAGSVM) etc. SVM with a linear Kernel is called linear SVM and one with a non-linear Kernel is called non-linear SVM. Linear SVM is an efficient technique for high dimensional data applications like document classification, word-sense disambiguation, drug design etc. because under such data applications, test accuracy of linear SVM is closer to non-linear SVM while its training is much faster than non-linear SVM. SVM is continuously evolving since its inception and researchers have proposed many problem formulations, solvers and strategies for solving SVM. Moreover, due to advancements in the technology, data has taken the form of ‘Big Data’ which have posed a challenge for Machine Learning to train a classifier on this large-scale data. In this paper, we have presented a review on evolution of linear support vector machine classification, its solvers, strategies to improve solvers, experimental results, current challenges and research directions.

We present a randomized algorithm that on input a symmetric, weakly diagonally dominant$n$ -by- $n$matrix$A$with$m$nonzero entries and an$n$ -vector$b$produces an${\\tilde{x}} $such that$\\|{\\tilde{x}} - A^{\\dagger} {b} \\|_{A} \\leq \\epsilon \\|A^{\\dagger} {b}\\|_{A}$in expected time$O (m \\log^{c}n \\log (1/\\epsilon))$for some constant$c$ . By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix$A$with nonpositive off-diagonal entries and$k \\geq 1$ , we construct in time$O (m \\log^{c} n)$a preconditioner$B$of$A$with at most$2 (n - 1) + O ((m/k) \\log^{39} n)$nonzero off-diagonal entries such that the finite generalized condition number$\\kappa_{f} (A,B)$is at most$k$ , for some other constant$c$ . In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time$O (n \\log^{2} n + n \\log n \\ \\log \\log n \\ \\log (1/\\epsilon))$ . We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice. [PUBLICATION ABSTRACT]

Sudoku is an NP-complete problem therefore developing various efficient algorithms is crucial. This paper presents an enhanced approach to solving sudoku puzzles by improving on recursive backtracking with constraint propagation and bitmask, focusing on the implementation of the naked pair technique. This research aims to add constraints to reduce the backtracking steps in solving sudoku using naked pairs, which is a technique that eliminates candidates of a cell from other cells in the same row, column or sub-grid when two cells share the same candidate sets. This research uses a dataset of 1 million sudoku puzzlers from Kaggle to evaluate the proposed solver’s performance. The solver first processes the puzzle into bitmask vectors then uses the singles and naked pairs technique to minimize the candidates, and then uses depth-first search to backtrack and solve the puzzle. As a result, the naked pair strategy slightly increases the total steps, and it significantly reduces the computation of depth-first search steps, making the solver potentially more efficient in solving difficult puzzles.

This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues. Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.

A review of preconditioned solver for large-scale applications in science and industry is presented. The analysis, parallelization, and optimization approach for large unstructured sparse matrices using IDR methods are considered for modern multicore microprocessors. Performance criteria and methods have been revisited, along with consideration of parallelization involving the solver and preconditioner using the OpenMP environment. Results of the successful implementation for efficient parallelization are presented.

The multi-probe anechoic chamber (MPAC) method is utilized to reconstruct the channel by distributing proper weights to the probes in the over-the-air (OTA) testing scenario. The commonly used probe weighting method is a convex optimization solver. However, the solver is time-consuming. Owing to the numerous probes required for massive multiple-input multiple-output (MIMO) OTA testing, more time will be taken. Two efficient probe weighting methods are proposed in this paper. The results of the simulation verify that the proposed methods have higher computing efficiency and can achieve accuracy similar to that of the convex optimization solver.

Dependent type theory is the foundation of many modern proof assistants. Inhabitation and unification are undecidable problems that are useful for theorem proving and program synthesis. We introduce Canonical-min, a sound and complete solver for inhabitation and unification in dependent type theory, implemented in 185 lines of Lean code. This paper describes a novel implementation of dependent type theory and a monadic framework to transform the type checker into a performant solver. Finally, we introduce DTTBench, a benchmark for type inhabitation in dependent type theory.

Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems, each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B, it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver-evaluator gap, maintaining a stronger correct-wrong separation even on instances where the underlying solver often fails to solve.

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of H 2 -matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of H or H 2 matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses H 2 -matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size 1000 λ × 1000 λ (for which N = 100 M ) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration.