Experimental study of random projections below the JL limit

Random projection is a method used to reduce dimensionality of desired objects with pair-wise distances preserved at a relatively high probability. The mathematical theory behind this is called the Johnson-Lindenstrauss (JL) lemma. So, the basic idea of the JL lemma is that a set of points in a high dimensional space p are randomly projected down to a lower dimensional space q. This q can be as low as q0 to still make sure that with a certain probability the projected pair-wise distances are within [plus-minus][epsilon], of the pairwise distances before the projection, where plus or minus [eplison] is usually a very small value. This technique has already been used in a variety of areas like clustering, image and text data processing. Lots of researchers have already studied the properties and performance of the JL lemma above q0 (q0 is usually called the JL limit or JL bound), where q = p-1, p-2,..., q0, but no research has investigated using the JL lemma below the JL limit (q = q0-1, q0-2,..., 2). With much lower dimension, the data processing, storing almost everything is going to be so much easier. We can visualize the clustering information about data sets in 2D plots. One thing should not be forgotten is that the distance preservation is probabilistic. How well will the distances being preserved below the JL bound? Will it affect or even completely destroy the cluster structure after the projection? What is a good projection method? We are going to study and answer these questions as much as we can in this thesis

Ergodic SDEs on submanifolds and related numerical sampling schemes

In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure {\mu} on the level set of a smooth function $\xi: \mathbb{R}^d\rightarrow \mathbb{R}^k$, $1\le k < d$. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is {\mu}. Motivated by the previous work of Ciccotti, Leli\`evre, and Vanden-Eijnden [11], as well as the work of Leli\`evre, Rousset, and Stoltz [33], in this paper we construct a family of ergodic diffusion processes on the level set of $\xi$ whose invariant measures coincide with the given one. For the conditional measure, in particular, we show that the corresponding SDEs of the constructed ergodic processes have relatively simple forms, and, moreover, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn't require computing the second derivatives of $\xi$ and the error estimates of its long time sampling efficiency are obtained.Comment: 45 pages. Accepted versio

Strong Orthogonal Decompositions and Nonlinear Impulse Response Functions for Infinite-Variance Processes

In this paper we prove Wold-type decompositions with strongorthogonal prediction innovations exist in smooth, re‡exive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. Our theory includes as special cases all previous Wold-type decompositions of discrete time processes; completely characterizes when nonlinear heavy-tailed processes obtain a strong-orthogonal moving average representation; and easily promotes a theory of nonlinear impulse response functions for infinite variance processes. We exemplify our theory by developing a nonlinear impulse response func tion for smooth transition threshold processes, we discuss how to test de composition innovations for strong orthogonality and whether the proposed model represents the best predictor, and we apply the methodology to currency exchange rates.Orthogonal decompositions, Banach spaces, projection iterations, infinite variance, moving average, nonlinear impulse response function, smooth transition autoregression, Lp-metric projection, Lp-GMM.

Quantum risk-sensitive estimation and robustness

This paper studies a quantum risk-sensitive estimation problem and investigates robustness properties of the filter. This is a direct extension to the quantum case of analogous classical results. All investigations are based on a discrete approximation model of the quantum system under consideration. This allows us to study the problem in a simple mathematical setting. We close the paper with some examples that demonstrate the robustness of the risk-sensitive estimator.Comment: 24 page

Limit theorems for the hierarchy of freeness

The central limit theorem, the invariance principle and the Poisson limit theorem for the hierarchy of freeness are studied. We show that for given natural m the limit laws can be expressed in terms of non-crossing partitions of depth smaller or equal to m. For the algebra of polynomials in one variable we solve the associated moment problems and find explicitly the discrete limit measures.Comment: 19 pages, latex, no figure