Zhi-Ming Ma

Introduction to the theory of (non-symmetric) Dirichlet forms

0 Introduction.- I Functional Analytic Background.- 1 Resolvents, semigroups, generators.- 2 Coercive bilinear forms.- 3 Closability.- 4 Contraction properties.- 5 Notes/References.- II Examples.- 1 Starting point: operator.- 2 Starting point: bilinear form - finite dimensional case.- 3 Starting point: bilinear form - infinite dimensional case.- 4 Starting point: semigroup of kernels.- 5 Starting point: resolvent of kernels.- 6 Notes/References.- III Analytic Potential Theory of Dirichlet Forms.- 1 Excessive functions and balayage.- 2 ?-exceptional sets and capacities.- 3 Quasi-continuity.- 4 Notes/References.- IV Markov Processes and Dirichlet Forms.- 1 Basics on Markov processes.- 2 Association of right processes and Dirichlet forms.- 3 Quasi-regularity and the construction of the process.- 4 Examples of quasi-regular Dirichlet forms.- 5 Necessity of quasi-regularity and some probabilistic potential theory.- 6 One-to-one correspondences.- 7 Notes/References.- V Characterization of Particular Processes.- 1 Local property and diffusions.- 2 A new capacity and Hunt processes.- 3 Notes/References.- VI Regularization.- 1 Local compactification.- 2 Consequences - the transfer method.- 3 Notes/References.- A Some Complements.- 1 Adjoint operators.- 2 The Banach/Alaoglu and Banach/Saks theorems.- 3 Supplement on Ray resolvents and right processes.

Supervised rank aggregation

This paper is concerned with rank aggregation, the task of combining the ranking results of individual rankers at meta-search. Previously, rank aggregation was performed mainly by means of unsupervised learning. To further enhance ranking accuracies, we propose employing supervised learning to perform the task, using labeled data. We refer to the approach as Supervised Rank Aggregation. We set up a general framework for conducting Supervised Rank Aggregation, in which learning is formalized an optimization which minimizes disagreements between ranking results and the labeled data. As case study, we focus on Markov Chain based rank aggregation in this paper. The optimization for Markov Chain based methods is not a convex optimization problem, however, and thus is hard to solve. We prove that we can transform the optimization problem into that of Semidefinite Programming and solve it efficiently. Experimental results on meta-searches show that Supervised Rank Aggregation can significantly outperform existing unsupervised methods.

BOUNDARY PROBLEMS FOR FRACTIONAL LAPLACIANS

By making use of the reflected α-stable process on a closed domain of ℝn and its killed subprocess on part of the domain, in this paper we study the boundary value problem for the Schrodinger type equation of a fractional Laplacian. The boundary condition is imposed partly follow Dirichlet condition and partly follow Neuman condition. We obtain the existence and the uniqueness resutls. The solution is expressed as a functional of the reflected α-stable process.

Markov processes associated with semi-Dirichlet forms

Recently, in [1], [11] an analytic characterization of all (non-symmetric) Dirichlet forms (on general state spaces) which are associated with pairs of special standard porocesses has been proved extending fundamental results in [8], [9], [18], [5], [10] (cf. also the literature in [11]). These Dirichlet forms are called quasi-regular (cf. Section 3 below). The processes forming the pairs are in duality w.r.t. the reference (speed) measure of the Dirichlet form. From a probabilistic point of view, however, this duality is quite restrictive. It arises from the fact that a Dirichlet form by the definition in [1], [11] exhibits a contraction property in both of its arguments. More precisely, we recall that a coercive closed form (<̂ ,Z>((ί)) on L(E;m) (cf. Section 2 below) is called a Dirichlet form if for all ueD(£) we have w MeD(S) and

Feynman-Kac semigroups in terms of signed smooth measures

We study positive continuous additive functionals of diffusion processes associated with Dirichlet forms. We also give necessary and sufficient conditions for a Feynman-Kac semigroup associated with a signed Borel measure to be a strongly continuous semigroup in the relevant L 2-space. We characterize their generators as second order partial differential operators with zero order term (potential) given by a signed measure (which can be so singular as to be nowhere Radon). We also discuss preservation of p-boundedness and L p -smoothing properties of semigroups under perturbations. We also study integral kernels of Feynman-Kac semigroups and provide upper bounds for them.

Generalization analysis of listwise learning-to-rank algorithms

This paper presents a theoretical framework for ranking, and demonstrates how to perform generalization analysis of listwise ranking algorithms using the framework. Many learning-to-rank algorithms have been proposed in recent years. Among them, the listwise approach has shown higher empirical ranking performance when compared to the other approaches. However, there is no theoretical study on the listwise approach as far as we know. In this paper, we propose a theoretical framework for ranking, which can naturally describe various listwise learning-to-rank algorithms. With this framework, we prove a theorem which gives a generalization bound of a listwise ranking algorithm, on the basis of Rademacher Average of the class of compound functions. The compound functions take listwise loss functions as outer functions and ranking models as inner functions. We then compute the Rademacher Averages for existing listwise algorithms of ListMLE, ListNet, and RankCosine. We also discuss the tightness of the bounds in different situations with regard to the list length and transformation function.

Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms

The theory of regular Dirichlet forms (E,3ί) associated with a locally compact separable metric space 3C and a positive Radon measure m s.t. s\ιρp[m]=3£ is a well developed subject, both from the potential analytic and the probabilistic point of view. It has its origins in work by Beurling-Deny and was particularly pursued by Fukushima and Silverstein see e.g. [19], [27] and references therein. It presents, at least in the symmetric case, a natural extension of the continuous functions framework of classical and axiomatic potential theory in the functional analytical (IΛfunctions) direction, covering in particular a stochastic calculus for generators with coefficients which are not restricted to be functions (the associated processes need not be semimartingales). This theory has turned out, in the last 15 years, to be particularly suited for applications in quantum theory, see e.g. [4], [5], [20], [1], [8]. In this field, but also in other contexts, see e.g. [2], there is the necessity of studying certain generalized functional of the processes (of Feynman-Kac type), corresponding to singular perturbations of a given Dirichlet form (e.g. the one associated with the Laplacian over R). This is discussed e.g. in [2], [10], [28], [29], [30], [16], [3], [1], [IS], [11], [12], [23], [22] and references therein. Many of the discussions have been concerned with functional associated with measures in the so called Kato class (cfr. [9], [26]). They are particular cases of smooth measures (in the sense of [19]) for the Dirichlet form associated with the Laplacian. It is natural to ask oneselves what happens if one tries to carry through similar constructions using an arbitrary smooth measure associated with a general (regular) Dirichlet form. In the present paper we initiate such a study. We give results on the structure

Query-level stability and generalization in learning to rank

This paper is concerned with the generalization ability of learning to rank algorithms for information retrieval (IR). We point out that the key for addressing the learning problem is to look at it from the viewpoint of query. We define a number of new concepts, including query-level loss, query-level risk, and query-level stability. We then analyze the generalization ability of learning to rank algorithms by giving query-level generalization bounds to them using query-level stability as a tool. Such an analysis is very helpful for us to derive more advanced algorithms for IR. We apply the proposed theory to the existing algorithms of Ranking SVM and IRSVM. Experimental results on the two algorithms verify the correctness of the theoretical analysis.

A framework to compute page importance based on user behaviors

This paper is concerned with a framework to compute the importance of webpages by using real browsing behaviors of Web users. In contrast, many previous approaches like PageRank compute page importance through the use of the hyperlink graph of the Web. Recently, people have realized that the hyperlink graph is incomplete and inaccurate as a data source for determining page importance, and proposed using the real behaviors of Web users instead. In this paper, we propose a formal framework to compute page importance from user behavior data (which covers some previous works as special cases). First, we use a stochastic process to model the browsing behaviors of Web users. According to the analysis on hundreds of millions of real records of user behaviors, we justify that the process is actually a continuous-time time-homogeneous Markov process, and its stationary probability distribution can be used as the measure of page importance. Second, we propose a number of ways to estimate parameters of the stochastic process from real data, which result in a group of algorithms for page importance computation (all referred to as BrowseRank). Our experimental results have shown that the proposed algorithms can outperform the baseline methods such as PageRank and TrustRank in several tasks, demonstrating the advantage of using our proposed framework.

AggregateRank: bringing order to web sites

Since the website is one of the most important organizational structures of the Web, how to effectively rank websites has been essential to many Web applications, such as Web search and crawling. In order to get the ranks of websites, researchers used to describe the inter-connectivity among websites with a so-called HostGraph in which the nodes denote websites and the edges denote linkages between websites (if and only if there are hyperlinks from the pages in one website to the pages in the other, there will be an edge between these two websites), and then adopted the random walk model in the HostGraph. However, as pointed in this paper, the random walk over such a HostGraph is not reasonable because it is not in accordance with the browsing behavior of web surfers. Therefore, the derivate rank cannot represent the true probability of visiting the corresponding website.In this work, we mathematically proved that the probability of visiting a website by the random web surfer should be equal to the sum of the PageRank values of the pages inside that website. Nevertheless, since the number of web pages is much larger than that of websites, it is not feasible to base the calculation of the ranks of websites on the calculation of PageRank. To tackle this problem, we proposed a novel method named AggregateRank rooted in the theory of stochastic complement, which cannot only approximate the sum of PageRank accurately, but also have a lower computational complexity than PageRank. Both theoretical analysis and experimental evaluation show that AggregateRank is a better method for ranking websites than previous methods.