Related papers: A Modification of LLR
The problem of quickest change detection in a sequence of independent observations is considered. The pre-change distribution is assumed to be known, while the post-change distribution is unknown. Two tests based on post-change density…
Research on nearest-neighbor methods tends to focus somewhat dichotomously either on the statistical or the computational aspects -- either on, say, Bayes consistency and rates of convergence or on techniques for speeding up the proximity…
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which…
Non-negative least squares (NNLS) problem is one of the most important fundamental problems in numeric analysis. It has been widely used in scientific computation and data modeling. In big data, the limitations of algorithm speed and…
The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published.…
The Lempel-Ziv parsing of a string (LZ77 for short) is one of the most important and widely-used algorithmic tools in data compression and string processing. We show that the Lempel-Ziv parsing of a string of length $n$ on an alphabet of…
A factor model with a break in its factor loadings is observationally equivalent to a model without changes in the loadings but a change in the variance of its factors. This effectively transforms a structural change problem of high…
The Dickman function $\rho(u)$ gives the asymptotic probability that a large integer $N$ has no prime divisor exceeding $N^{1/u}$. We expand it in terms of rapidly computable multiple polylogarithms, as defined by Goncharov and intensively…
The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/\epsilon))$ when solving up to $\epsilon$ relative error, is a long-standing open problem in numerical linear algebra…
In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic…
Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by…
Dirichlet's proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Euler's earlier work on the…
Primal heuristics play a crucial role in quickly finding feasible solutions for NP-hard integer linear programming (ILP). Although $\textit{end-to-end learning}$-based primal heuristics (E2EPH) have recently been proposed, they are…
This is an expository paper intended to introduce the polynomial time lattice basis reduction algorithm first described by Arjen Lenstra, Hendrik Lenstra, and L\'aszl\'o Lov\'asz in 1982. We begin by introducing the shortest vector problem,…
We analyze algorithms for computing the $n$th prime $p_n$ and establish asymptotic bounds for several approaches. Using existing results on the complexity of evaluating the prime-counting function $\pi(x)$, we show that the binary search…
In this paper we consider two different views of the model checking problems for the Linear Temporal Logic (LTL). On the one hand, we consider the universal model checking problem for LTL, where one asks that for a given system and a given…
We consider the Pm || Cmax scheduling problem where the goal is to schedule n jobs on m identical parallel machines to minimize makespan. We revisit the famous Longest Processing Time (LPT) rule proposed by Graham in 1969. LPT requires to…
In this paper we provide a provably convergent algorithm for the multivariate Gaussian Maximum Likelihood version of the Behrens--Fisher Problem. Our work builds upon a formulation of the log-likelihood function proposed by Buot and…
Given a zero-dimensional ideal I in K[x1,...,xn] of degree D, the transformation of the ordering of its Groebner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving…
We present TransNormerLLM, the first linear attention-based Large Language Model (LLM) that outperforms conventional softmax attention-based models in terms of both accuracy and efficiency. TransNormerLLM evolves from the previous linear…