This paper studies linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.

We define infinitesimal gradient boosting as a limit of the popular tree-based gradient boosting algorithm from machine learning. The limit is considered in the vanishing-learning-rate asymptotic, that is when the learning rate tends to zero and the number of gradient trees is rescaled accordingly. For this purpose, we introduce a new class of randomized regression trees bridging totally randomized trees and Extra Trees and using a softmax distribution for binary splitting. Our main result is the convergence of the associated stochastic algorithm and the characterization of the limiting procedure as the unique solution of a nonlinear ordinary differential equation in a infinite dimensional function space. Infinitesimal gradient boosting defines a smooth path in the space of continuous functions along which the training error decreases, the residuals remain centered and the total variation is well controlled.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

Remark: the vector $x$ defined above can be viewed as a $n\times1$ matrix and is more particularly called a column-vector. Remark: for matrices $A,B$, we have $(AB) T B TA T$. Remark: not all square matrices are invertible. Remark: $A$ is invertible if and only if $ A eq0$. Linearly dependence A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others.

This article provides an overview of stochastic process and fundamental mathematical concepts that are important to understand. Stochastic variable is a variable that moves in random order. Exchange rates, interest rates or stock prices are stochastic in nature. Stochastic variables can follow wiener or Itos process. I will start by explaining what stochastic process is.

This book introduces probability, statistics and stochastic processes to students. It can be used by both students and practitioners in engineering, various sciences, finance, and other related fields. It provides a clear and intuitive approach to these topics while maintaining mathematical accuracy. You can also find courses and videos online.

Sample space The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$. Event Any subset $E$ of the sample space is known as an event. That is, an event is a set consisting of possible outcomes of the experiment. If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. Axiom 1 ― Every probability is between 0 and 1 included, i.e: Axiom 2 ― The probability that at least one of the elementary events in the entire sample space will occur is 1, i.e: Axiom 3 ― For any sequence of mutually exclusive events $E_1, ..., E_n$, we have: Permutation A permutation is an arrangement of $r$ objects from a pool of $n$ objects, in a given order.

Background: Non-target screening consists in searching a sample for all present substances, suspected or unknown, with very little prior knowledge about the sample. This approach has been introduced more than a decade ago in the field of water analysis, but is still very scarce for indoor and atmospheric trace gas measurements, despite the clear need for a better understanding of the atmospheric trace gas composition. For a systematic detection of emerging trace gases in the atmosphere, a new and powerful analytical method is gas chromatography (GC) of preconcentrated samples, followed by electron ionisation, high resolution mass spectrometry (EI-HRMS). In this work, we present data analysis tools to enable automated identification of unknown compounds measured by GC-EI-HRMS. Results: Based on co-eluting mass/charge fragments, we developed an innovative data analysis method to reliably reconstruct the chemical formulae of the fragments, using efficient combinatorics and graph theory. The method (i) does not to require the presence of the molecular ion, which is absent in $\sim$40% of EI spectra, and (ii) permits to use all measured data while giving more weight to mass/charge ratios measured with better precision. Our method has been trained and validated on >50 halocarbons and hydrocarbons with a molar masses of 30-330 g mol-1 , measured with a mass resolution of approx. 3500. For >90% of the compounds, more than 90% of the reconstructed signal is correct. Cases of wrong identification can be attributed to the scarcity of detected fragments per compound (less than six measured mass/charge) or the lack of isotopic constrain (no rare isotopocule detected). Conclusions: Our method enables to reconstruct most probable chemical formulae independently from spectral databases. Therefore, it demonstrates the suitability of EI-HRMS data for non-target analysis and paves the way for the identification of substances for which no EI mass spectrum is registered in databases. We illustrate the performances of our method for atmospheric trace gases and suggest that it may be well suited for many other types of samples.

Machine Reassignment is a challenging problem for constraint programming (CP) and mixed-integer linear programming (MILP) approaches, especially given the size of data centres. The multi-objective version of the Machine Reassignment Problem is even more challenging and it seems unlikely for CP or MILP to obtain good results in this context. As a result, the first approaches to address this problem have been based on other optimisation methods, including metaheuristics. In this paper we study under which conditions a mixed-integer optimisation solver, such as IBM ILOG CPLEX, can be used for the Multi-objective Machine Reassignment Problem. We show that it is useful only for small or medium-scale data centres and with some relaxations, such as an optimality tolerance gap and a limited number of directions explored in the search space. Building on this study, we also investigate a hybrid approach, feeding a metaheuristic with the results of CPLEX, and we show that the gains are important in terms of quality of the set of Pareto solutions (+126.9% against the metaheuristic alone and +17.8% against CPLEX alone) and number of solutions (8.9 times more than CPLEX), while the processing time increases only by 6% in comparison to CPLEX for execution times larger than 100 seconds.

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming.[1] It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices.[citation The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory; though the basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior and Abraham Wald's sequential analysis.[citation In continuous-time optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation.[4][5] In discrete time any multi-stage optimization problem can be solved by analyzing the appropriate Bellman equation.