Choosing a diverse subset of a large collection of points in a metric space is a fundamental problem, with applications in feature selection, recommender systems, web search, data summarization, etc. Various notions of diversity have been proposed, tailored to different applications. The general algorithmic goal is to find a subset of points that maximize diversity, while obeying a cardinality (or more generally, matroid) constraint. The goal of this paper is to develop a novel linear programming (LP) framework that allows us to design approximation algorithms for such problems. We study an objective known as {\em sum-min} diversity, which is known to be effective in many applications, and give the first constant factor approximation algorithm. Our LP framework allows us to easily incorporate additional constraints, as well as secondary objectives. We also prove a hardness result for two natural diversity objectives, under the so-called {\em planted clique} assumption. Finally, we study the empirical performance of our algorithm on several standard datasets. We first study the approximation quality of the algorithm by comparing with the LP objective. Then, we compare the quality of the solutions produced by our method with other popular diversity maximization algorithms.

Estimating optimal phylogenetic trees or hierarchical clustering trees from metric data is an important problem in evolutionary biology and data analysis. Intuitively, the goodness-of-fit of a metric space to a tree depends on its inherent treeness, as well as other metric properties such as intrinsic dimension. Existing algorithms for embedding metric spaces into tree metrics provide distortion bounds depending on cardinality. Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric. We consider an embedding of a metric space into a tree proposed by Gromov. By proving a stability result, we obtain an improved additive distortion bound depending only on the hyperbolicity and doubling dimension of the metric. We observe that Gromov's method is dual to the well-known single linkage hierarchical clustering (SLHC) method. By means of this duality, we are able to transport our results to the setting of SLHC, where such additive distortion bounds were previously unknown.

We propose a pool-based non-parametric active learning algorithm for general metric spaces, called MArgin Regularized Metric Active Nearest Neighbor (MARMANN), which outputs a nearest-neighbor classifier. We give prediction error guarantees that depend on the noisy-margin properties of the input sample, and are competitive with those obtained by previously proposed passive learners. We prove that the label complexity of MARMANN is significantly lower than that of any passive learner with similar error guarantees. Our algorithm is based on a generalized sample compression scheme and a new label-efficient active model-selection procedure.

We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

Many machine learning problems, especially multi-modal learning problems, have two sets of distinct features (e.g., image and text features in news story classification, or neuroimaging data and neurocognitive data in cognitive science research). This paper addresses the joint dimensionality reduction of two feature vectors in supervised learning problems. In particular, we assume a discriminative model where low-dimensional linear embeddings of the two feature vectors are sufficient statistics for predicting a dependent variable. We show that a simple algorithm involving singular value decomposition can accurately estimate the embeddings provided that certain sample complexities are satisfied, without specifying the nonlinear link function (regressor or classifier). The main results establish sample complexities under multiple settings. Sample complexities for different link functions only differ by constant factors.

A key barrier to making phonetic studies scalable and replicable is the need to rely on subjective, manual annotation. To help meet this challenge, a machine learning algorithm was developed for automatic measurement of a widely used phonetic measure: vowel duration. Manually-annotated data were used to train a model that takes as input an arbitrary length segment of the acoustic signal containing a single vowel that is preceded and followed by consonants and outputs the duration of the vowel. The model is based on the structured prediction framework. The input signal and a hypothesized set of a vowel's onset and offset are mapped to an abstract vector space by a set of acoustic feature functions. The learning algorithm is trained in this space to minimize the difference in expectations between predicted and manually-measured vowel durations. The trained model can then automatically estimate vowel durations without phonetic or orthographic transcription. Results comparing the model to three sets of manually annotated data suggest it out-performed the current gold standard for duration measurement, an HMM-based forced aligner (which requires orthographic or phonetic transcription as an input).

Back in 1970, the social activist Irina Dunn scribbled a slogan on the back of a toilet cubicle door at the University of Sydney. It said: "A woman needs a man like a fish needs a bicycle." The phrase went viral and eventually became a famous refrain for the growing feminist movement of the time. The phrase is also an example of sarcasm. The humor comes from the fact that a fish doesn't need a bicycle.

Multilingual Text Classification (MLTC) is a text classification task in which documents are written each in one among a set L of natural languages, and in which all documents must be classified under the same classification scheme, irrespective of language. There are two main variants of MLTC, namely Cross-Lingual Text Classification (CLTC) and Polylingual Text Classification (PLTC). In PLTC, which is the focus of this paper, we assume (differently from CLTC) that for each language in L there is a representative set of training documents; PLTC consists of improving the accuracy of each of the |L| monolingual classifiers by also leveraging the training documents written in the other (|L| − 1) languages. The obvious solution, consisting of generating a single polylingual classifier from the juxtaposed monolingual vector spaces, is usually infeasible, since the dimensionality of the resulting vector space is roughly |L| times that of a monolingual one, and is thus often unmanageable. As a response, the use of machine translation tools or multilingual dictionaries has been proposed. However, these resources are not always available, or are not always free to use. One machine-translation-free and dictionary-free method that, to the best of our knowledge, has never been applied to PLTC before, is Random Indexing (RI). We analyse RI in terms of space and time efficiency, and propose a particular configuration of it (that we dub Lightweight Random Indexing LRI). By running experiments on two well known public benchmarks, Reuters RCV1/RCV2 (a comparable corpus) and JRC-Acquis (a parallel one), we show LRI to outperform (both in terms of effectiveness and efficiency) a number of previously proposed machine-translation-free and dictionary-free PLTC methods that we use as baselines.

Last time we investigated the k-nearest-neighbors algorithm and the underlying idea that one can learn a classification rule by copying the known classification of nearby data points. This required that we view our data as sitting inside a metric space; that is, we imposed a kind of geometric structure on our data. One glaring problem is that there may be no reasonable way to do this. While we mentioned scaling issues and provided a number of possible metrics in our primer, a more common problem is that the data simply isn't numeric. For instance, a poll of US citizens might ask the respondent to select which of a number of issues he cares most about. There could be 50 choices, and there is no reasonable way to assign these numerical values so that all are equidistant in the resulting metric space. Another issue is that the quality of the data could be bad. For instance, there may be missing values for some attributes (e.g., a respondent may neglect to answer one or more questions).

I hope this post is okay (if not let me know). I'm attaching a pdf which rigorously defines my question. Briefly, what I'm wondering is this - for the set of data points {x1,...,xp} in a vector space, (say, Rn) under what conditions is the set {k(x1,),...,k(xp,)} (where k(,) is a kernel function) independent? What conditions must the set {x1,...,xp} and the kernel function have to ensure independence? If there isn't an immediate answer to this question I'll happily take recommendations for mathematical reading towards trying to answer this question.