Shengjun Pan

The maximum likelihood probability of unique-singleton, ternary, and length-7 patterns

We derive several pattern maximum likelihood (PML) results, among them showing that if a pattern has only one symbol appearing once, its PML support size is at most twice the number of distinct symbols, and that if the pattern is ternary with at most one symbol appearing once, its PML support size is three. We apply these results to extend the set of patterns whose PML distribution is known to all ternary patterns, and to all but one pattern of length up to seven.

Adaptive non-local means for multiview image denoising: Searching for the right patches via a statistical approach

We present an adaptive non-local means (NLM) denoising method for a sequence of images captured by a multiview imaging system, where direct extensions of existing single image NLM methods are incapable of producing good results. Our proposed method consists of three major components: (1) a robust joint-view distance metric to measure the similarity of patches; (2) an adaptive procedure derived from statistical properties of the estimates to determine the optimal number of patches to be used; (3) a new NLM algorithm to denoise using only a set of similar patches. Experimental results show that the proposed method is robust to disparity estimation error, out-performs existing algorithms in multiview settings, and performs competitively in video settings.

The maximum likelihood probability of skewed patterns

A pattern is skewed if, as in 11123, one of its symbols repeats and the others appear once. We show that the pattern-maximum-likelihood distribution of essentially all skewed patterns consists of one discrete element whose probability is the fraction of times the repeated symbol appears in the pattern.

Recent results on pattern maximum likelihood

We derive some general sufficient conditions for the uniformity of the Pattern Maximum Likelihood distribution (PML). We also provide upper bounds on the support size of a class of patterns, and mention some recent results about the PML of 1112234.

On reconstructing a string from its substring compositions

Motivated by protein sequencing, we consider the problem of reconstructing a string from the compositions of its substrings. We provide several results, including the following. General classes of strings that cannot be distinguished from their substring compositions. An almost complete characterization of the lengths for which reconstruction is possible. Bounds on the number of strings with the same substring compositions in terms of the number of divisors of the string length plus one. A relation to the turnpike problem and a bivariate polynomial formulation of string reconstruction.

Exact calculation of pattern probabilities

We describe two algorithms for calculating the probability of m-symbol length-n patterns over k-element distributions, a partition-based algorithm with complexity roughly 2O(m log m) and a recursive algorithm with complexity roughly 2O(m+log n) with the precise bounds provided in the text. The problem is related to symmetric-polynomial evaluation, and the analysis reveals a connection to the number of connected graphs.

Classification using pattern probability estimators

We consider the problem of classification, where the data of the classes are generated i.i.d. according to unknown probability distributions. The goal is to classify test data with minimum error probability, based on the training data available for the classes. The Likelihood Ratio Test (LRT) is the optimal decision rule when the distributions are known. Hence, a popular approach for classification is to estimate the likelihoods using well known probability estimators, e.g., the Laplace and Good-Turing estimators, and use them in a LRT. We are primarily interested in situations where the alphabet of the underlying distributions is large compared to the training data available, which is indeed the case in most practical applications. We motivate and propose LRTs based on pattern probability estimators that are known to achieve low redundancy for universal compression of large alphabet sources. While a complete proof for optimality of these decision rules is warranted, we demonstrate their performance and compare it with other well-known classifiers by various experiments on synthetic data and real data for text classification.

Algebraic computation of pattern maximum likelihood

Pattern maximum likelihood (PML) is a technique for estimating the probability multiset of an unknown distribution. With any random sample, it associates the distribution maximizing the probability of its pattern. The required computation is a maximization of a monomial symmetric polynomial over the monotone simplex. The PML of only very few patterns have been found analytically, and for other patterns, the PML has been approximated by a heuristic algorithm. Taking an algebraic approach, we determine the PML of short patterns by solving a system of multivariate polynomial equations using the method of resultants. Using this approach, we determine the PML of the pattern 1112234, the last length-7 pattern whose PML was unknown. Under two plausible but yet unproved assumptions on the optimal alphabet size and the number of distinct probabilities, we also find the PML distribution of all previously unknown patterns of length up to 14.

On the query computation and verification of functions

In the query model of multi-variate function computation, the values of the variables are queried sequentially, in an order that may depend on previously revealed values, until the functions value can be determined. The functions computation query complexity is the lowest expected number of queries required by any query order. Instead of computation, it is often easier to consider verification, where the value of the function is given and the queries aim to verify it. The lowest expected number of queries necessary is the functions verification query complexity. We show that for all symmetric functions of independent binary random variables, the computation and verification complexities coincide. This provides a simple method for finding the query complexity and the optimal query order for computing many functions. We also show that if the symmetry condition is removed, there are functions whose verification complexity is strictly lower than their computation complexity, and mention that the same holds when the independence or binary conditions are removed.