Sanjoy Dasgupta

Robi Bhattacharjee, Sanjoy Dasgupta, Kamalika Chaudhuri

There has been some recent interest in detecting and addressing memorization of training data by deep neural networks. A formal framework for memorization in generative models, called "data-copying," was proposed by Meehan et. al. (2020). We build upon their work to show that their framework may fail to detect certain kinds of blatant memorization. Motivated by this and the theory of non-parametric methods, we provide an alternative definition of data-copying that applies more locally. We provide a method to detect data-copying, and provably show that it works with high probability when enough data is available. We also provide lower bounds that characterize the sample requirement for reliable detection.

Anthony Thomas, Behnam Khaleghi, Gopi Krishna Jha et al.

Hyperdimensional computing (HDC) is a paradigm for data representation and learning originating in computational neuroscience. HDC represents data as high-dimensional, low-precision vectors which can be used for a variety of information processing tasks like learning or recall. The mapping to high-dimensional space is a fundamental problem in HDC, and existing methods encounter scalability issues when the input data itself is high-dimensional. In this work, we explore a family of streaming encoding techniques based on hashing. We show formally that these methods enjoy comparable guarantees on performance for learning applications while being substantially more efficient than existing alternatives. We validate these results experimentally on a popular high-dimensional classification problem and show that our approach easily scales to very large data sets.

Sanjoy Dasgupta, Geelon So

We study an instance of online non-parametric classification in the realizable setting. In particular, we consider the classical 1-nearest neighbor algorithm, and show that it achieves sublinear regret - that is, a vanishing mistake rate - against dominated or smoothed adversaries in the realizable setting.

Sanjoy Dasgupta, Yoav Freund

We present a general-purpose active learning scheme for data in metric spaces. The algorithm maintains a collection of neighborhoods of different sizes and uses label queries to identify those that have a strong bias towards one particular label; when two such neighborhoods intersect and have different labels, the region of overlap is treated as a ``known unknown'' and is a target of future active queries. We give label complexity bounds for this method that do not rely on assumptions about the data and we instantiate them in several cases of interest.

Marco Bazzani, Sanjoy Dasgupta

The $k$-d tree is one of the oldest and most widely used data structures for nearest neighbor search. It partitions Euclidean space into axis-aligned rectangular cells. There are two standard ways to find the nearest neighbor to a query in a $k$-d tree. Defeatist search returns the closest data point in the query's cell, while comprehensive search also searches other cells as needed to guarantee it finds the nearest neighbor. Both strategies are commonly believed to perform poorly in high dimensions, but there have been few theoretical results explaining this. We prove non-asymptotic bounds on the runtime of comprehensive search and the accuracy of defeatist search. Under mild distributional assumptions, when the dimension $d$ is at least polylogarithmic in the number of data points, defeatist search is no more likely to return the nearest neighbor than random guessing, and comprehensive search visits every cell with high probability. We also show that on uniform data, with high probability, comprehensive search visits at most $2^{\mathcal{O}(d)}$ cells when each cell contains at least logarithmically many data points, and defeatist search returns the nearest neighbor when each cell additionally contains at least $2^{\mathcal{O}(d \log d)}$ data points. Finally, for arbitrary absolutely continuous distributions, we upper bound the expected distance between the query and the point returned by defeatist search.

Casey Meehan, Kamalika Chaudhuri, Sanjoy Dasgupta

Detecting overfitting in generative models is an important challenge in machine learning. In this work, we formalize a form of overfitting that we call {\em{data-copying}} -- where the generative model memorizes and outputs training samples or small variations thereof. We provide a three sample non-parametric test for detecting data-copying that uses the training set, a separate sample from the target distribution, and a generated sample from the model, and study the performance of our test on several canonical models and datasets. For code \& examples, visit https://github.com/casey-meehan/data-copying

Nicolas A. Errandonea, Santiago Mazuelas, Jose A. Lozano et al.

Prior work on partial labels learning (PLL) has shown that learning is possible even when each instance is associated with a bag of labels, rather than a single accurate but costly label. However, the necessary conditions for learning with partial labels remain unclear, and existing PLL methods are effective only in specific scenarios. In this work, we mathematically characterize the settings in which PLL is feasible. In addition, we present PL A-$k$NN, an adaptive nearest-neighbors algorithm for PLL that is effective in general scenarios and enjoys strong performance guarantees. Experimental results corroborate that PL A-$k$NN can outperform state-of-the-art methods in general PLL scenarios.

Robert R. Nerem, Samantha Chen, Sanjoy Dasgupta et al.

Neural networks (NNs), despite their success and wide adoption, still struggle to extrapolate out-of-distribution (OOD), i.e., to inputs that are not well-represented by their training dataset. Addressing the OOD generalization gap is crucial when models are deployed in environments significantly different from the training set, such as applying Graph Neural Networks (GNNs) trained on small graphs to large, real-world graphs. One promising approach for achieving robust OOD generalization is the framework of neural algorithmic alignment, which incorporates ideas from classical algorithms by designing neural architectures that resemble specific algorithmic paradigms (e.g. dynamic programming). The hope is that trained models of this form would have superior OOD capabilities, in much the same way that classical algorithms work for all instances. We rigorously analyze the role of algorithmic alignment in achieving OOD generalization, focusing on graph neural networks (GNNs) applied to the canonical shortest path problem. We prove that GNNs, trained to minimize a sparsity-regularized loss over a small set of shortest path instances, exactly implement the Bellman-Ford (BF) algorithm for shortest paths. In fact, if a GNN minimizes this loss within an error of $ε$, it implements the BF algorithm with an error of $O(ε)$. Consequently, despite limited training data, these GNNs are guaranteed to extrapolate to arbitrary shortest-path problems, including instances of any size. Our empirical results support our theory by showing that NNs trained by gradient descent are able to minimize this loss and extrapolate in practice.

Sanjoy Dasgupta, Syamantak Kumar, Shourya Pandey et al.

Low-precision streaming PCA estimates the top principal component in a streaming setting under limited precision. We establish an information-theoretic lower bound on the quantization resolution required to achieve a target accuracy for the leading eigenvector. We study Oja's algorithm for streaming PCA under linear and nonlinear stochastic quantization. The quantized variants use unbiased stochastic quantization of the weight vector and the updates. Under mild moment and spectral-gap assumptions on the data distribution, we show that a batched version achieves the lower bound up to logarithmic factors under both schemes. This leads to a nearly dimension-free quantization error in the nonlinear quantization setting. Empirical evaluations on synthetic streams validate our theoretical findings and demonstrate that our low-precision methods closely track the performance of standard Oja's algorithm.

Verónica Álvarez, Santiago Mazuelas, Steven An et al.

The accurate labeling of datasets is often both costly and time-consuming. Given an unlabeled dataset, programmatic weak supervision obtains probabilistic predictions for the labels by leveraging multiple weak labeling functions (LFs) that provide rough guesses for labels. Weak LFs commonly provide guesses with assorted types and unknown interdependences that can result in unreliable predictions. Furthermore, existing techniques for programmatic weak supervision cannot provide assessments for the reliability of the probabilistic predictions for labels. This paper presents a methodology for programmatic weak supervision that can provide confidence intervals for label probabilities and obtain more reliable predictions. In particular, the methods proposed use uncertainty sets of distributions that encapsulate the information provided by LFs with unrestricted behavior and typology. Experiments on multiple benchmark datasets show the improvement of the presented methods over the state-of-the-art and the practicality of the confidence intervals presented.

Akash Kumar, Sanjoy Dasgupta

In this work, we investigate the problem of learning distance functions within the query-based learning framework, where a learner is able to pose triplet queries of the form: ``Is $x_i$ closer to $x_j$ or $x_k$?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: $ω$-additive approximation and $(1 + ω)$-multiplicative approximation. For the additive approximation, we propose a global method whose query complexity is quadratic in the size of a finite cover of the sample space. For the (stronger) multiplicative approximation, we introduce a method that combines global and local approaches, utilizing multiple Mahalanobis distance functions to capture local geometry. This method has a query complexity that scales quadratically with both the size of the cover and the ambient space dimension of the sample space.

Sanjoy Dasgupta, Geelon So

In the realizable online setting, a learner is tasked with making predictions for a stream of instances, where the correct answer is revealed after each prediction. A learning rule is online consistent if its mistake rate eventually vanishes. The nearest neighbor rule (Fix and Hodges, 1951) is a fundamental prediction strategy, but it is only known to be consistent under strong statistical or geometric assumptions: the instances come i.i.d. or the label classes are well-separated. We prove online consistency for all measurable functions in doubling metric spaces under the mild assumption that the instances are generated by a process that is uniformly absolutely continuous with respect to a finite, upper doubling measure.

Sanjoy Dasgupta, Eduardo Laber

Linkage methods are among the most popular algorithms for hierarchical clustering. Despite their relevance the current knowledge regarding the quality of the clustering produced by these methods is limited. Here, we improve the currently available bounds on the maximum diameter of the clustering obtained by complete-link for metric spaces. One of our new bounds, in contrast to the existing ones, allows us to separate complete-link from single-link in terms of approximation for the diameter, which corroborates the common perception that the former is more suitable than the latter when the goal is producing compact clusters. We also show that our techniques can be employed to derive upper bounds on the cohesion of a class of linkage methods that includes the quite popular average-link.

Sanjoy Dasgupta, Gaurav Mahajan, Geelon So

We prove asymptotic convergence for a general class of $k$-means algorithms performed over streaming data from a distribution: the centers asymptotically converge to the set of stationary points of the $k$-means cost function. To do so, we show that online $k$-means over a distribution can be interpreted as stochastic gradient descent with a stochastic learning rate schedule. Then, we prove convergence by extending techniques used in optimization literature to handle settings where center-specific learning rates may depend on the past trajectory of the centers.

Sanjoy Dasgupta, Nave Frost, Michal Moshkovitz

We study the faithfulness of an explanation system to the underlying prediction model. We show that this can be captured by two properties, consistency and sufficiency, and introduce quantitative measures of the extent to which these hold. Interestingly, these measures depend on the test-time data distribution. For a variety of existing explanation systems, such as anchors, we analytically study these quantities. We also provide estimators and sample complexity bounds for empirically determining the faithfulness of black-box explanation systems. Finally, we experimentally validate the new properties and estimators.

Yang Shen, Sanjoy Dasgupta, Saket Navlakha

Continual learning in computational systems is challenging due to catastrophic forgetting. We discovered a two layer neural circuit in the fruit fly olfactory system that addresses this challenge by uniquely combining sparse coding and associative learning. In the first layer, odors are encoded using sparse, high dimensional representations, which reduces memory interference by activating non overlapping populations of neurons for different odors. In the second layer, only the synapses between odor activated neurons and the output neuron associated with the odor are modified during learning; the rest of the weights are frozen to prevent unrelated memories from being overwritten. We show empirically and analytically that this simple and lightweight algorithm significantly boosts continual learning performance. The fly associative learning algorithm is strikingly similar to the classic perceptron learning algorithm, albeit two modifications, which we show are critical for reducing catastrophic forgetting. Overall, fruit flies evolved an efficient lifelong learning algorithm, and circuit mechanisms from neuroscience can be translated to improve machine computation.

Robi Bhattacharjee, Jacob Imola, Michal Moshkovitz et al.

We consider online $k$-means clustering where each new point is assigned to the nearest cluster center, after which the algorithm may update its centers. The loss incurred is the sum of squared distances from new points to their assigned cluster centers. The goal over a data stream $X$ is to achieve loss that is a constant factor of $L(X, OPT_k)$, the best possible loss using $k$ fixed points in hindsight. We propose a data parameter, $Λ(X)$, such that for any algorithm maintaining $O(k\text{poly}(\log n))$ centers at time $n$, there exists a data stream $X$ for which a loss of $Ω(Λ(X))$ is inevitable. We then give a randomized algorithm that achieves clustering loss $O(Λ(X) + L(X, OPT_k))$. Our algorithm uses $O(k\text{poly}(\log n))$ memory and maintains $O(k\text{poly}(\log n))$ cluster centers. Our algorithm also enjoys a running time of $O(k\text{poly}(\log n))$ and is the first algorithm to achieve polynomial space and time complexity in this setting. It also is the first to have provable guarantees without making any assumptions on the input data.

Anthony Thomas, Sanjoy Dasgupta, Tajana Rosing

Hyperdimensional (HD) computing is a set of neurally inspired methods for obtaining high-dimensional, low-precision, distributed representations of data. These representations can be combined with simple, neurally plausible algorithms to effect a variety of information processing tasks. HD computing has recently garnered significant interest from the computer hardware community as an energy-efficient, low-latency, and noise-robust tool for solving learning problems. In this review, we present a unified treatment of the theoretical foundations of HD computing with a focus on the suitability of representations for learning.

Sanjoy Dasgupta, Christopher Tosh

A simple sparse coding mechanism appears in the sensory systems of several organisms: to a coarse approximation, an input $x \in \R^d$ is mapped to much higher dimension $m \gg d$ by a random linear transformation, and is then sparsified by a winner-take-all process in which only the positions of the top $k$ values are retained, yielding a $k$-sparse vector $z \in \{0,1\}^m$. We study the benefits of this representation for subsequent learning. We first show a universal approximation property, that arbitrary continuous functions of $x$ are well approximated by linear functions of $z$, provided $m$ is large enough. This can be interpreted as saying that $z$ unpacks the information in $x$ and makes it more readily accessible. The linear functions can be specified explicitly and are easy to learn, and we give bounds on how large $m$ needs to be as a function of the input dimension $d$ and the smoothness of the target function. Next, we consider whether the representation is adaptive to manifold structure in the input space. This is highly dependent on the specific method of sparsification: we show that adaptivity is not obtained under the winner-take-all mechanism, but does hold under a slight variant. Finally we consider mappings to the representation space that are random but are attuned to the data distribution, and we give favorable approximation bounds in this setting.

Sanjoy Dasgupta, Sivan Sabato

Recent work introduced the model of learning from discriminative feature feedback, in which a human annotator not only provides labels of instances, but also identifies discriminative features that highlight important differences between pairs of instances. It was shown that such feedback can be conducive to learning, and makes it possible to efficiently learn some concept classes that would otherwise be intractable. However, these results all relied upon perfect annotator feedback. In this paper, we introduce a more realistic, robust version of the framework, in which the annotator is allowed to make mistakes. We show how such errors can be handled algorithmically, in both an adversarial and a stochastic setting. In particular, we derive regret bounds in both settings that, as in the case of a perfect annotator, are independent of the number of features. We show that this result cannot be obtained by a naive reduction from the robust setting to the non-robust setting.

Sanjoy Dasgupta, Nave Frost, Michal Moshkovitz et al.

Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a complicated way. To improve interpretability, we consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We study this problem from a theoretical viewpoint, measuring cluster quality by the $k$-means and $k$-medians objectives: Must there exist a tree-induced clustering whose cost is comparable to that of the best unconstrained clustering, and if so, how can it be found? In terms of negative results, we show, first, that popular top-down decision tree algorithms may lead to clusterings with arbitrarily large cost, and second, that any tree-induced clustering must in general incur an $Ω(\log k)$ approximation factor compared to the optimal clustering. On the positive side, we design an efficient algorithm that produces explainable clusters using a tree with $k$ leaves. For two means/medians, we show that a single threshold cut suffices to achieve a constant factor approximation, and we give nearly-matching lower bounds. For general $k \geq 2$, our algorithm is an $O(k)$ approximation to the optimal $k$-medians and an $O(k^2)$ approximation to the optimal $k$-means. Prior to our work, no algorithms were known with provable guarantees independent of dimension and input size.

Sanjoy Dasgupta, Stefanos Poulis, Christopher Tosh

The formalism of anchor words has enabled the development of fast topic modeling algorithms with provable guarantees. In this paper, we introduce a protocol that allows users to interact with anchor words to build customized and interpretable topic models. Experimental evidence validating the usefulness of our approach is also presented.

Akshay Balsubramani, Sanjoy Dasgupta, Yoav Freund et al.

We introduce a variant of the $k$-nearest neighbor classifier in which $k$ is chosen adaptively for each query, rather than supplied as a parameter. The choice of $k$ depends on properties of each neighborhood, and therefore may significantly vary between different points. (For example, the algorithm will use larger $k$ for predicting the labels of points in noisy regions.) We provide theory and experiments that demonstrate that the algorithm performs comparably to, and sometimes better than, $k$-NN with an optimal choice of $k$. In particular, we derive bounds on the convergence rates of our classifier that depend on a local quantity we call the `advantage' which is significantly weaker than the Lipschitz conditions used in previous convergence rate proofs. These generalization bounds hinge on a variant of the seminal Uniform Convergence Theorem due to Vapnik and Chervonenkis; this variant concerns conditional probabilities and may be of independent interest.

Robi Bhattacharjee, Sanjoy Dasgupta

We consider the problem of embedding a relation, represented as a directed graph, into Euclidean space. For three types of embeddings motivated by the recent literature on knowledge graphs, we obtain characterizations of which relations they are able to capture, as well as bounds on the minimal dimensionality and precision needed.

Christopher Tosh, Sanjoy Dasgupta

In this work, we describe a framework that unifies many different interactive learning tasks. We present a generalization of the {\it query-by-committee} active learning algorithm for this setting, and we study its consistency and rate of convergence, both theoretically and empirically, with and without noise.

Ehsan Kazemi, Lin Chen, Sanjoy Dasgupta et al.

There is increasing interest in learning algorithms that involve interaction between human and machine. Comparison-based queries are among the most natural ways to get feedback from humans. A challenge in designing comparison-based interactive learning algorithms is coping with noisy answers. The most common fix is to submit a query several times, but this is not applicable in many situations due to its prohibitive cost and due to the unrealistic assumption of independent noise in different repetitions of the same query. In this paper, we introduce a new weak oracle model, where a non-malicious user responds to a pairwise comparison query only when she is quite sure about the answer. This model is able to mimic the behavior of a human in noise-prone regions. We also consider the application of this weak oracle model to the problem of content search (a variant of the nearest neighbor search problem) through comparisons. More specifically, we aim at devising efficient algorithms to locate a target object in a database equipped with a dissimilarity metric via invocation of the weak comparison oracle. We propose two algorithms termed WORCS-I and WORCS-II (Weak-Oracle Comparison-based Search), which provably locate the target object in a number of comparisons close to the entropy of the target distribution. While WORCS-I provides better theoretical guarantees, WORCS-II is applicable to more technically challenging scenarios where the algorithm has limited access to the ranking dissimilarity between objects. A series of experiments validate the performance of our proposed algorithms.

Sanjoy Dasgupta, Michael Luby

We introduce a new model of interactive learning in which an expert examines the predictions of a learner and partially fixes them if they are wrong. Although this kind of feedback is not i.i.d., we show statistical generalization bounds on the quality of the learned model.

Christopher Tosh, Sanjoy Dasgupta

To date, the tightest upper and lower-bounds for the active learning of general concept classes have been in terms of a parameter of the learning problem called the splitting index. We provide, for the first time, an efficient algorithm that is able to realize this upper bound, and we empirically demonstrate its good performance.

Sharad Vikram, Sanjoy Dasgupta

Clustering is a powerful tool in data analysis, but it is often difficult to find a grouping that aligns with a user's needs. To address this, several methods incorporate constraints obtained from users into clustering algorithms, but unfortunately do not apply to hierarchical clustering. We design an interactive Bayesian algorithm that incorporates user interaction into hierarchical clustering while still utilizing the geometry of the data by sampling a constrained posterior distribution over hierarchies. We also suggest several ways to intelligently query a user. The algorithm, along with the querying schemes, shows promising results on real data.

Sanjoy Dasgupta

The development of algorithms for hierarchical clustering has been hampered by a shortage of precise objective functions. To help address this situation, we introduce a simple cost function on hierarchies over a set of points, given pairwise similarities between those points. We show that this criterion behaves sensibly in canonical instances and that it admits a top-down construction procedure with a provably good approximation ratio.

Akshay Balsubramani, Sanjoy Dasgupta, Yoav Freund

We consider a situation in which we see samples in $\mathbb{R}^d$ drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both.

Kamalika Chaudhuri, Sanjoy Dasgupta

Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor classification has not fully reflected these subtle properties. We analyze the behavior of these estimators in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. As a by-product, we are able to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previously known. We illustrate our upper and lower bounds by introducing smoothness classes that are customized for nearest neighbor classification.

Margareta Ackerman, Sanjoy Dasgupta

The explosion in the amount of data available for analysis often necessitates a transition from batch to incremental clustering methods, which process one element at a time and typically store only a small subset of the data. In this paper, we initiate the formal analysis of incremental clustering methods focusing on the types of cluster structure that they are able to detect. We find that the incremental setting is strictly weaker than the batch model, proving that a fundamental class of cluster structures that can readily be detected in the batch setting is impossible to identify using any incremental method. Furthermore, we show how the limitations of incremental clustering can be overcome by allowing additional clusters.

Kamalika Chaudhuri, Sanjoy Dasgupta, Samory Kpotufe et al.

For a density $f$ on ${\mathbb R}^d$, a {\it high-density cluster} is any connected component of $\{x: f(x) \geq λ\}$, for some $λ> 0$. The set of all high-density clusters forms a hierarchy called the {\it cluster tree} of $f$. We present two procedures for estimating the cluster tree given samples from $f$. The first is a robust variant of the single linkage algorithm for hierarchical clustering. The second is based on the $k$-nearest neighbor graph of the samples. We give finite-sample convergence rates for these algorithms which also imply consistency, and we derive lower bounds on the sample complexity of cluster tree estimation. Finally, we study a tree pruning procedure that guarantees, under milder conditions than usual, to remove clusters that are spurious while recovering those that are salient.

Matus Telgarsky, Sanjoy Dasgupta

Suppose $k$ centers are fit to $m$ points by heuristically minimizing the $k$-means cost; what is the corresponding fit over the source distribution? This question is resolved here for distributions with $p\geq 4$ bounded moments; in particular, the difference between the sample cost and distribution cost decays with $m$ and $p$ as $m^{\min\{-1/4, -1/2+2/p\}}$. The essential technical contribution is a mechanism to uniformly control deviations in the face of unbounded parameter sets, cost functions, and source distributions. To further demonstrate this mechanism, a soft clustering variant of $k$-means cost is also considered, namely the log likelihood of a Gaussian mixture, subject to the constraint that all covariance matrices have bounded spectrum. Lastly, a rate with refined constants is provided for $k$-means instances possessing some cluster structure.

Sanjoy Dasgupta

We consider the task of learning the maximum-likelihood polytree from data. Our first result is a performance guarantee establishing that the optimal branching (or Chow-Liu tree), which can be computed very easily, constitutes a good approximation to the best polytree. We then show that it is not possible to do very much better, since the learning problem is NP-hard even to approximately solve within some constant factor.

Sanjoy Dasgupta, Leonard Schulman

Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chosen model will be used in the future for prediction tasks involving more ``focused' predictive distributions. Although focused predictive distributions can be produced from the joint probability distribution by marginalization, in practice the best model in the unsupervised sense does not necessarily perform well in supervised domains. In particular, the standard marginal likelihood score is a criterion for the unsupervised task, and, although frequently used for supervised model selection also, does not perform well in such tasks. In this paper we study the performance of the marginal likelihood score empirically in supervised Bayesian network selection tasks by using a large number of publicly available classification data sets, and compare the results to those obtained by alternative model selection criteria, including empirical crossvalidation methods, an approximation of a supervised marginal likelihood measure, and a supervised version of Dawids prequential(predictive sequential) principle.The results demonstrate that the marginal likelihood score does NOT perform well FOR supervised model selection, WHILE the best results are obtained BY using Dawids prequential r napproach.

Sanjoy Dasgupta

Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic and real data.

Matus Telgarsky, Sanjoy Dasgupta

This manuscript develops the theory of agglomerative clustering with Bregman divergences. Geometric smoothing techniques are developed to deal with degenerate clusters. To allow for cluster models based on exponential families with overcomplete representations, Bregman divergences are developed for nondifferentiable convex functions.

Sanjoy Dasgupta, Daniel Hsu, Nakul Verma

X in R^D has mean zero and finite second moments. We show that there is a precise sense in which almost all linear projections of X into R^d (for d < D) look like a scale-mixture of spherical Gaussians -- specifically, a mixture of distributions N(0, sigma^2 I_d) where the weight of the particular sigma component is P (| X |^2 = sigma^2 D). The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X's distribution. We explore this result in a variety of experiments.

Nakul Verma, Samory Kpotufe, Sanjoy Dasgupta

Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees, dyadic trees, and PCA trees. Our motivation is to get a feel for (i) the kind of intrinsic low dimensional structure that can be empirically verified, (ii) the extent to which a spatial partition can exploit such structure, and (iii) the implications for standard statistical tasks such as regression, vector quantization, and nearest neighbor search.