In this technical report, we discuss several sampling algorithms for Determinantal Point Processes (DPP). DPPs have recently gained a broad interest in the machine learning and statistics literature as random point processes with negative correlation, i.e., ones that can generate a"diverse"sample from a set of items. They are parametrized by a matrix $\mathbf{L}$, called $L$-ensemble, that enco...

We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar’s interior-point algorithm in [DST02], we show that the smoothed complexity of an interior-point algorithm for linear programming is O(m log(m/σ)). In contrast, the best known bound on the worst-case complexity of linear programming is O(mL), wher...

Floating-point numbers are limited both in range and in precision, yet they are widely used as a way to implement computations on real numbers. Thus arithmetic operations introduce small errors which might be amplified during subsequent computations and cause inaccuracies. As such, proving the correctness of a floating-point algorithm usually entails verifying that the computed results are stil...

In this paper, we proposed algorithms interior proximal methods based on entropylike distance for the minimization of the quasiconvex function subjected to nonnegativity constraints. Under the assumptions that the objective function is bounded below and continuously differentiable, we established the well definedness of the sequence generated by the algorithms and obtained two important converg...

We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solver over polytopes, thereby partially answering a 30 year-old conjecture. We a...

A set of n points in the plane is in equiangular configuration if there exist a center and an ordering of the points such that the angle of each two adjacent points w.r.t. the center is 360 ◦ n , i.e., if all angles between adjacent points are equal. We show that there is at most one center of equiangularity, and we give a linear time algorithm that decides whether a given point set is in equia...

As very powerful and important tools in the study of nonlinear sciences, fixed point methods have attracted so much attention. Over the last decades, fixed point techniques have been applied extensively in such diverse fields as biology, chemistry, economics, engineering, game theory, physics, and so on. The thorough study of the fixed point theory and its approach methods contained in the lite...

A fast and accurate quantization noise estimator aiming at fixed-point implementations of Digital Signal Processing (DSP) algorithms is presented. The estimator enables significant reduction in the computation time required to perform complex wordlength optimizations. The proposed estimator is based on the use of Affine Arithmetic (AA) and it is presented in two versions: (i) a general version ...

Several point-set pattern-discovery and compression algorithms designed for analysing music are reviewed and evaluated. Each algorithm takes as input a point-set representation of a score in which each note is represented as a point in pitch-time space. Each algorithm computes the maximal translatable patterns (MTPs) in this input and the translational equivalence classes (TECs) of these MTPs, ...