Clustered Integer 3SUM via Additive Combinatorics

We present a collection of new results on problems related to 3SUM, including: 1. The first truly subquadratic algorithm for $\ \ \ \ \$ 1a. computing the (min,+) convolution for monotone increasing sequences with integer values bounded by $O(n)$, $\ \ \ \ \$1b. solving 3SUM for monotone sets in 2D with integer coordinates bounded by $O(n)$, and $\ \ \ \ \$1c. preprocessing a binary string for histogram indexing (also called jumbled indexing). The running time is: $O(n^{(9+\sqrt{177})/12}\,\textrm{polylog}\,n)=O(n^{1.859})$ with randomization, or $O(n^{1.864})$ deterministically. This greatly improves the previous $n^2/2^{\Omega(\sqrt{\log n})}$ time bound obtained from Williams' recent result on all-pairs shortest paths [STOC'14], and answers an open question raised by several researchers studying the histogram indexing problem. 2. The first algorithm for histogram indexing for any constant alphabet size that achieves truly subquadratic preprocessing time and truly sublinear query time. 3. A truly subquadratic algorithm for integer 3SUM in the case when the given set can be partitioned into $n^{1-\delta}$ clusters each covered by an interval of length $n$, for any constant $\delta>0$. 4. An algorithm to preprocess any set of $n$ integers so that subsequently 3SUM on any given subset can be solved in $O(n^{13/7}\,\textrm{polylog}\,n)$ time. All these results are obtained by a surprising new technique, based on the Balog--Szemer\'edi--Gowers Theorem from additive combinatorics

A Bayesian alternative to mutual information for the hierarchical clustering of dependent random variables

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering

Integrated Inference and Learning of Neural Factors in Structural Support Vector Machines

Tackling pattern recognition problems in areas such as computer vision, bioinformatics, speech or text recognition is often done best by taking into account task-specific statistical relations between output variables. In structured prediction, this internal structure is used to predict multiple outputs simultaneously, leading to more accurate and coherent predictions. Structural support vector machines (SSVMs) are nonprobabilistic models that optimize a joint input-output function through margin-based learning. Because SSVMs generally disregard the interplay between unary and interaction factors during the training phase, final parameters are suboptimal. Moreover, its factors are often restricted to linear combinations of input features, limiting its generalization power. To improve prediction accuracy, this paper proposes: (i) Joint inference and learning by integration of back-propagation and loss-augmented inference in SSVM subgradient descent; (ii) Extending SSVM factors to neural networks that form highly nonlinear functions of input features. Image segmentation benchmark results demonstrate improvements over conventional SSVM training methods in terms of accuracy, highlighting the feasibility of end-to-end SSVM training with neural factors

Segregating Event Streams and Noise with a Markov Renewal Process Model

DS and MP are supported by EPSRC Leadership Fellowship EP/G007144/1