Cardinality sketches are compact data structures for representing sets or vectors. These sketches are space-efficient, typically requiring only logarithmic storage in the input size, and enable approximation of cardinality (or the number of nonzero entries). A crucial property in applications is composability of the sketching map, meaning that the sketch of a union of sets can be computed from individual sketches. Existing designs provide strong statistical guarantees, ensuring that a randomly sampled sketching map is accurate with high probability for a number of queries that is exponential in the sketch size \(k\). However, these guarantees degrade to quadratic in \(k\) when queries are adaptive, meaning they depend on previous responses.

We propose a new class of linear Transformers called FourierLearner-Transformers (FLTs), which incorporate a wide range of relative positional encoding mechanisms (RPEs). These include regular RPE techniques applied for sequential data, as well as novel RPEs operating on geometric data embedded in higher-dimensional Euclidean spaces. FLTs construct the optimal RPE mechanism implicitly by learning its spectral representation. As opposed to other architectures combining efficient low-rank linear attention with RPEs, FLTs remain practical in terms of their memory usage and do not require additional assumptions about the structure of the RPE mask. Besides, FLTs allow for applying certain structural inductive bias techniques to specify masking strategies, e.g. they provide a way to learn the so-called local RPEs introduced in this paper and give accuracy gains as compared with several other linear Transformers for language modeling. We also thoroughly test FLTs on other data modalities and tasks, such as image classification, 3D molecular modeling, and learnable optimizers. To the best of our knowledge, for 3D molecular data, FLTs are the first Transformer architectures providing linear attention and incorporating RPE masking.

One of the most basic problems for studying the “price of privacy over time” is the so called {\em private counter problem}, introduced by Dwork et al. (2010) and Chan et al. (2011). In this problem, we aim to track the number of {\em events} that occur over time, while hiding the existence of every single event. More specifically, in every time step $t\in[T]$ we learn (in an online fashion) that $\Delta_t\geq 0$ new events have occurred, and must respond with an estimate $n_t\approx\sum_{j=1}^t \Delta_j$. The privacy requirement is that {\em all of the outputs together}, across all time steps, satisfy {\em event level} differential privacy. The main question here is how our error needs to depend on the total number of time steps $T$ and the total number of events $n$. Dwork et al. (2015) showed an upper bound of $O\left(\log(T)+\log^2(n)\right)$, and Henzinger et al. (2023) showed a lower bound of $\Omega\left( \min\{\log n, \log T\} \right)$. We show a new lower bound of $\Omega\left(\min\{n,\log T\}\right)$, which is tight w.r.t. the dependence on $T$, and is tight in the sparse case where $\log^2 n=O(\log T)$. Our lower bound has the following implications: \begin{itemize} \item We show that our lower bound extends to the {\em online thresholds} problem, where the goal is to privately answer many “quantile queries” when these queries are presented one-by-one. This resolves an open question of Bun et al. (2017). \item Our lower bound implies, for the first time, a separation between the number of mistakes obtainable by a private online learner and a non-private online learner. This partially resolves a COLT’22 open question published by Sanyal and Ramponi. \item Our lower bound also yields the first separation between the standard model of private online learning and a recently proposed relaxed variant of it, called {\em private online prediction}. \end{itemize}

We present Self-Adaptive Robust Attention for Robotics Transformers (SARA-RT): a new paradigm for addressing the emerging challenge of scaling up Robotics Transformers (RT) for on-robot deployment. SARA-RT relies on the new method of fine-tuning proposed by us, called up-training. It converts pre-trained or already fine-tuned Transformer-based robotic policies of quadratic time complexity (including massive billion-parameter vision-language-action models or VLAs), into their efficient linear-attention counterparts maintaining high quality. We demonstrate the effectiveness of SARA-RT by speeding up: (a) the class of recently introduced RT-2 models [1], the first VLA robotic policies pre-trained on internet-scale data, as well as (b) Point Cloud Transformer (PCT) robotic policies operating on large point clouds. We complement our results with the rigorous mathematical analysis providing deeper insight into the phenomenon of SARA.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular *low displacement rank*) for integrating tensor fields defined on weighted trees. Several applications of the resulting *fast tree-field integrators* (FTFIs) are presented, including: (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) *Topological Transformers* (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as **three** extra learnable parameters per Transformer layer, leading to **1.0-1.5\%+** accuracy gains. Importantly, most of FTFIs are **exact** methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide **5.7-13x** speedups. We also provide an extensive theoretical analysis of our methods.

CountSketch and Feature Hashing (the ``hashing trick'') are popular randomized dimensionality reduction methods that support recovery of l2 -heavy hitters and approximate inner products. When the inputs are not adaptive (do not depend on prior outputs), classic estimators applied to a sketch of size O(l / epsilon) are accurate for a number of queries that is exponential in l. When inputs are adaptive, however, an adversarial input can be constructed after O(l) queries with the classic estimator and the best known robust estimator only supports ~O(l^2) queries. In this work we show that this quadratic dependence is in a sense inherent: We design an attack that after O(l^2) queries produces an adversarial input vector whose sketch is highly biased. Our attack uses ``natural'' non-adaptive inputs (only the final adversarial input is chosen adaptively) and universally applies with any correct estimator, including one that is unknown to the attacker. In that, we expose inherent vulnerability of this fundamental method.

We present two new classes of algorithms for efficient field integration on graphs encoding point cloud data. The first class, $\mathrm{SeparatorFactorization}$ (SF), leverages the bounded genus of point cloud mesh graphs, while the second class, $\mathrm{RFDiffusion}$ (RFD), uses popular $\epsilon$-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g. shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (in particular for mesh-dynamics modeling) as well as Wasserstein distance computations for point clouds, including the Gromov-Wasserstein variant.

Composition theorems are general and powerful tools that facilitate privacy accounting across multiple data accesses from per-access privacy bounds. However they often result in weaker bounds compared with end-to-end analysis. Two popular tools that mitigate that are the exponential mechanism (or report noisy max) and the sparse vector technique, generalized in a recent private selection framework by Liu and Talwar (STOC 2019). In this work, we propose a flexible framework of private selection and testing that generalizes the one proposed by Liu and Talwar, supporting a wide range of applications. We apply our framework to solve several fundamental tasks, including query releasing, top-k selection, and stable selection, with improved confidence-accuracy tradeoffs. Additionally, for online settings, we apply our private testing to design a mechanism for adaptive query releasing, which improves the sample complexity dependence on the confidence parameter for the celebrated private multiplicative weights algorithm of Hardt and Rothblum (FOCS 2010).

The problem of efficient approximation of a linear operator induced by the Gaussian or softmax kernel is often addressed using random features (RFs) which yield an unbiased approximation of the operator's result. Such operators emerge in important applications ranging from kernel methods to efficient Transformers. We propose parameterized, positive, non-trigonometric RFs which approximate Gaussian and softmax-kernels. In contrast to traditional RF approximations, parameters of these new methods can be optimized to reduce the variance of the approximation, and the optimum can be expressed in closed form. We show that our methods lead to variance reduction in practice (e^{10}-times smaller variance and beyond) and outperform previous methods in a kernel regression task. Using our proposed mechanism, we also present FAVOR#, a method for self-attention approximation in Transformers. We show that FAVOR# outperforms other random feature methods in speech modelling and natural language processing.

Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting where we want to find a good rank $k$ approximation to $f(U \cdot V)$, where $U, V^\top \in \mathbb{R}^{n \times r}$ are given, $r = O(\log(n))$, and $f(x)$ is a general scalar function. Previous work in sublinear low rank approximation has shown that if both (1) $U = V^\top$ and (2) $f(x)$ is a PSD kernel function, then there is an $O(nk^{\omega-1})$ time constant relative error approximation algorithm, where $\omega \approx 2.376$ is the exponent of matrix multiplication. We give the first conditional time hardness results for this problem, demonstrating that both conditions (1) and (2) are in fact necessary for getting better than $n^{2-o(1)}$ time for a relative error low rank approximation for a wide class of functions. We give novel reductions from the Strong Exponential Time Hypothesis (SETH) that rely on lower bounding the leverage scores of flat sparse vectors and hold even when the rank of the transformed matrix $f(UV)$ and the target rank are $n^{o(1)}$, and when $U = V^\top$. Furthermore, even when $f(x) = x^p$ is a simple polynomial, we give runtime lower bounds in the case when $U \neq V^\top$ of the form $\Omega(\min(n^{2-o(1)}, \Omega(2^p)))$. Lastly, we demonstrate that our lower bounds are tight by giving an $O(n \cdot \text{poly}(k, 2^p, 1/\epsilon))$ time relative error approximation algorithm and a fast $O(n \cdot \text{poly}(k, p, 1/\epsilon))$ additive error approximation using fast tensor-based sketching. Additionally, since our low rank algorithms rely on matrix-vector product subroutines, our lower bounds extend to show that computing $f(UV)W$, for even a small matrix $W$, requires $\Omega(n^{2-o(1)})$ time.

The problem of learning threshold functions is a fundamental one in machine learning. Classical learning theory implies sample complexity of O(ξ−1 log(1/β)) (for generalization error ξ with confidence 1−β). The private version of the problem, however, is more challenging and in particular, the sample complexity must depend on the size |X| of the domain. Progress on quantifying this dependence, via lower and upper bounds, was made in a line of works over the past decade. In this paper, we finally close the gap for approximate-DP and provide a nearly tight upper bound of O(log* |X|), which matches a lower bound by Alon et al (that applies even with improper learning) and improves over a prior upper bound of O((log* |X|)1.5) by Kaplan et al. We also provide matching upper and lower bounds of Θ(2log*|X|) for the additive error of private quasi-concave optimization (a related and more general problem). Our improvement is achieved via the novel Reorder-Slice-Compute paradigm for private data analysis which we believe will have further applications.

In this paper we provide, to the best of our knowledge, the first comprehensive approach for incorporating various masking mechanisms into Transformers architectures in a scalable way. We show that recent results on linear causal attention (Choromanski et al., 2021) and log-linear RPE-attention (Luo et al., 2021) are special cases of this general mechanism. However by casting the problem as a topological (graph-based) modulation of unmasked attention, we obtain several results unknown before, including efficient d-dimensional RPE-masking and graph-kernel masking. We leverage many mathematical techniques ranging from spectral analysis through dynamic programming and random walks to new algorithms for solving Markov processes on graphs. We provide a corresponding empirical evaluation.

The last decade saw impressive progress towards understanding the performance of algorithms in adaptive settings, where subsequent inputs may depend on the output from prior inputs. Adaptive settings arise in processes with feedback or with adversarial attacks. Existing designs of robust algorithms are generic wrappers of non-robust counterparts and leave open the possibility of better tailored designs. The lowers bounds (attacks) are similarly worst-case and their significance to practical setting is unclear. Aiming to understand these questions, we study the robustness of \texttt{CountSketch}, a popular dimensionality reduction technique that maps vectors to a lower dimension using randomized linear measurements. The sketch supports recovering $\ell_2$-heavy hitters of a vector (entries with $v[i]^2 \geq \frac{1}{k}\|\boldsymbol{v}\|^2_2$). We show that the classic estimator is not robust, and can be attacked with a number of queries of the order of the sketch size. We propose a robust estimator (for a slightly modified sketch) that allows for quadratic number of queries in the sketch size, which is an improvement factor of $\sqrt{k}$ (for $k$ heavy hitters) over prior "blackbox" approaches.

We introduce chefs' random tables (CRTs), a new class of non-trigonometric random features (RFs) to approximate Gaussian and softmax kernels. CRTs are an alternative to standard random kitchen sink (RKS) methods, which inherently rely on the trigonometric maps. We present variants of CRTs where RFs are positive, a key requirement for applications in recent low-rank Transformers. Further variance reduction is possible by leveraging statistics which are simple to compute. One instantiation of CRTs, the optimal positive random features (OPRFs), is to our knowledge the first RF method for unbiased softmax kernel estimation with positive and bounded RFs, resulting in exponentially small tails and much lower variance than its counterparts. As we show, orthogonal random features applied in OPRFs provide additional variance reduction for any dimensionality $d$ (not only asymptotically for sufficiently large $d$, as for RKS). We test CRTs on many tasks ranging from non-parametric classification to training Transformers for text, speech and image data, obtaining new state-of-the-art results for low-rank text Transformers, while providing linear space and time complexity.

Common datasets have the form of elements with keys (e.g., transactions and products) and the goal is to perform analytics on the aggregated form of key and frequency pairs. A weighted sample of keys by (a function of) frequency is a highly versatile summary that provides a sparse set of representative keys and supports approximate evaluations of query statistics. We propose private weighted sampling (PWS): A method that sanitizes a weighted sample as to ensure element-level differential privacy, while retaining its utility to the maximum extent possible. PWS maximizes the reporting probabilities of keys and estimation quality of a broad family of statistics. PWS improves over the state of the art even for the well-studied special case of private histograms, when no sampling is performed. We empirically observe significant performance gains of 20%-300% increase in key reporting for common Zipfian frequency distributions and accurate estimation with x2-8 lower frequencies. PWS is applied as a post-processing of a non-private sample, without requiring the original data. Therefore, it can be a seamless addition to existing implementations, such as those optimizes for distributed or streamed data. We believe that due to practicality and performance, PWS may become a method of choice in applications where privacy is desired.

We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group O(d) and naturally reductive homogeneous manifolds obtained from the action of the rotation group SO(d). We theoretically and experimentally demonstrate that our methods can be applied in various fields of machine learning including deep, convolutional and recurrent neural networks, reinforcement learning, normalizing flows and metric learning. We show an intriguing connection between efficient stochastic optimization on the orthogonal group and graph theory (e.g. matching problem, partition functions over graphs, graph-coloring). We leverage the theory of Lie groups and provide theoretical results for the designed class of algorithms. We demonstrate broad applicability of our methods by showing strong performance on the seemingly unrelated tasks of learning world models to obtain stable policies for the most difficult Humanoid agent from OpenAI Gym and improving convolutional neural networks.

Wasserstein distances are increasingly used in a wide variety of applications in machine learning. Sliced Wasserstein distances form an important subclass which may be estimated efficiently through one-dimensional sorting operations. In this paper, we propose a new variant of sliced Wasserstein distance, study the use of orthogonal coupling in Monte Carlo estimation of Wasserstein distances and draw connections with stratified sampling, and evaluate our approaches experimentally in a range of large-scale experiments in generative modelling and reinforcement learning.

We provide an online convex optimization algorithm with regret that interpolates between the regret of an algorithm using an optimal preconditioning matrix and one using a diagonal preconditioning matrix. Our regret bound is never worse than that obtained by diagonal preconditioning, and in certain setting even surpasses that of algorithms with full-matrix preconditioning. Importantly, our algorithm runs in the same time and space complexity as online gradient descent. Along the way we incorporate new techniques that mildly streamline and improve logarithmic factors in prior regret analyses. We conclude by benchmarking our algorithm on synthetic data and deep learning tasks.

We revisit the classic randomized sketch of a tensor product of $q$ vectors $x_i\in\mathbb{R}^n$. The $i$-th coordinate $(Sx)_i$ of the sketch is equal to $\prod_{j = 1}^q \langle u^{i, j}, x^j \rangle / \sqrt{m}$, where $u^{i,j}$ are independent random sign vectors. Kar and Karnick (JMLR, 2012) show that if the sketching dimension $m = \Omega(\epsilon^{-2} C_{\Omega}^2 \log (1/\delta))$, where $C_{\Omega}$ is a certain property of the point set $\Omega$ one wants to sketch, then with probability $1-\delta$, $\|Sx\|_2 = (1\pm \epsilon)\|x\|_2$ for all $x\in\Omega$. However, in their analysis $C_{\Omega}^2$ can be as large as $\Theta(n^{2q})$, even for a set $\Omega$ of $O(1)$ vectors $x$. We give a new analysis of this sketch, providing nearly optimal bounds. Namely, we show an upper bound of $m = \Theta \left (\epsilon^{-2} \log(n/\delta) + \epsilon^{-1} \log^q(n/\delta) \right ),$ which by composing with CountSketch, can be improved to $\Theta(\epsilon^{-2}\log(1/(\delta \epsilon)) + \epsilon^{-1} \log^q (1/(\delta \epsilon))$. For the important case of $q = 2$ and $\delta = 1/\poly(n)$, this shows that $m = \Theta(\epsilon^{-2} \log(n) + \epsilon^{-1} \log^2(n))$, demonstrating that the $\epsilon^{-2}$ and $\log^2(n)$ terms do not multiply each other. We also show a nearly matching lower bound of $m = \Omega(\eps^{-2} \log(1/(\delta)) + \eps^{-1} \log^q(1/(\delta)))$. In a number of applications, one has $|\Omega| = \poly(n)$ and in this case our bounds are optimal up to a constant factor. This is the first high probability sketch for tensor products that has optimal sketch size and can be implemented in $m \cdot \sum_{i=1}^q \textrm{nnz}(x_i)$ time, where $\textrm{nnz}(x_i)$ is the number of non-zero entries of $x_i$. Lastly, we empirically compare our sketch to other sketches for tensor products, and give a novel application to compressing neural networks.

We present an in-depth examination of the effectiveness of radial basis function kernel (beyond Gaussian) estimators based on orthogonal random feature maps. We show that orthogonal estimators outperform state-of-the-art mechanisms that use iid sampling under weak conditions for tails of the associated Fourier distributions. We prove that for the case of many dimensions, the superiority of the orthogonal transform can be accurately measured by a property we define called the charm of the kernel, and that orthogonal random features provide optimal (in terms of mean squared error) kernel estimators. We provide the first theoretical results which explain why orthogonal random features outperform unstructured on downstream tasks such as kernel ridge regression by showing that orthogonal random features provide kernel algorithms with better spectral properties than the previous state-of-the-art. Our results enable practitioners more generally to estimate the benefits from applying orthogonal transforms.

Monte Carlo sampling in high-dimensional, low-sample settings is important in many machine learning tasks. We improve current methods for sampling in Euclidean spaces by avoiding independence, and instead consider ways to couple samples. We show fundamental connections to optimal transport theory, leading to novel sampling algorithms, and providing new theoretical grounding for existing strategies. We compare our new strategies against prior methods for improving sample efficiency, including QMC, by studying discrepancy. We explore our findings empirically, and observe benefits of our sampling schemes for reinforcement learning and generative modelling.

We study the problem of automatically and efficiently generating itineraries for users who are on vacation. We focus on the common case, wherein the trip duration is more than a single day. Previous efficient algorithms based on greedy heuristics suffer from two problems. First, the itineraries are often unbalanced, with excellent days visiting top attractions followed by days of exclusively lower-quality alternatives. Second, the trips often re-visit neighborhoods repeatedly in order to cover increasingly low-tier points of interest. Our primary technical contribution is an algorithm that addresses both these problems by maximizing the quality of the worst day. We give theoretical results showing that this algorithm»s competitive factor is within a factor two of the guarantee of the best available algorithm for a single day, across many variations of the problem. We also give detailed empirical evaluations using two distinct datasets:(a) anonymized Google historical visit data and(b) Foursquare public check-in data. We show first that the overall utility of our itineraries is almost identical to that of algorithms specifically designed to maximize total utility, while the utility of the worst day of our itineraries is roughly twice that obtained from other approaches. We then turn to evaluation based on human raters who score our itineraries only slightly below the itineraries created by human travel experts with deep knowledge of the area.

We consider an efficient computational framework for speeding up several machine learning algorithms with almost no loss of accuracy. The proposed framework relies on projections via structured matrices that we call Structured Spinners, which are formed as products of three structured matrix-blocks that incorporate rotations. The approach is highly generic, i.e. i) structured matrices under consideration can either be fully-randomized or learned, ii) our structured family contains as special cases all previously considered structured schemes, iii) the setting extends to the non-linear case where the projections are followed by non-linear functions, and iv) the method finds numerous applications including kernel approximations via random feature maps, dimensionality reduction algorithms,new fast cross-polytope LSH techniques, deep learning, convex optimization algorithms via Newton sketches, quantization with random projection trees, and more. The proposed framework comes with theoretical guarantees characterizing the capacity of the structured model in reference to its unstructured counterpart and is based on a general theoretical principle that we describe in the paper. As a consequence of our theoretical analysis, we provide the first theoretical guarantees for one of the most efficient existing LSH algorithms based on the HD 3 HD 2 HD 1 structured matrix [Andoni et al., 2015]. The exhaustive experimental evaluation confirms the accuracy and efficiency of structured spinners for a variety of different applications.

We introduce LAMP: the Linear Additive Markov Process. Transitions in LAMP may be influenced by states visited in the distant history of the process, but unlike higher-order Markov processes, LAMP retains an efficient parameterization. LAMP also allows the specific dependence on history to be learned efficiently from data.

Random walk is an important tool in many graph mining applications including estimating graph parameters, sampling portions of the graph, and extracting dense communities. In this paper we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using random walk as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting.

Networks are characterized by nodes and edges. While there has been a spate of recent work on estimating the number of nodes in a network, the edge-estimation question appears to be largely unaddressed. In this work we consider the problem of estimating the average degree of a large network using efficient random sampling, where the number of nodes is not known to the algorithm. We propose a new estimator for this problem that relies on access to node samples under a prescribed distribution. Next, we show how to efficiently realize this ideal estimator in a random walk setting. Our estimator has a natural and simple implementation using random walks; we bound its performance in terms of the mixing time of the underlying graph. We then show that our estimators are both provably and practically better than many natural estimators for the problem. Our work contrasts with existing theoretical work on estimating average degree, which assume that a uniform random sample of nodes is available and the number of nodes is known.

Inverted indexing is a ubiquitous technique used in retrieval systems including web search. Despite its popularity, it has a drawback - query retrieval time is highly variable and grows with the corpus size. In this work we propose an alternative technique, permutation indexing, where retrieval cost is strictly bounded and has only logarithmic dependence on the corpus size. Our approach is based on two novel techniques: (a) partitioning of the term space into overlapping clusters of terms that frequently co-occur in queries, and (b) a data structure for compactly encoding results of all queries composed of terms in a cluster as continuous sequences of document ids. Then, query results are retrieved by fetching few small chunks of these sequences. There is a price though: our encoding is lossy and thus returns approximate result sets. The fraction of the true results returned, recall, is controlled by the level of redundancy. The more space is allocated for the permutation index the higher is the recall. We analyze permutation indexing both theoretically under simplified document and query models, and empirically on a realistic document and query collections. We show that although permutation indexing can not replace traditional retrieval methods, since high recall cannot be guaranteed on all queries, it covers up to 77% of tail queries and can be used to speed up retrieval for these queries.

Fast nonlinear function classes are crucial for nonparametric estimation, such as in kernel methods. This paper proposes an improvement to random kitchen sinks that offers significantly faster computation in log-linear time without sacrificing accuracy. Furthermore, we show how one may adjust the regularization properties of the kernel simply by changing the spectral distribution of the projection matrix. We provide experimental results which show that even for for moderately small problems we already achieve two orders of magnitude faster computation and three orders of magnitude lower memory footprint.

In this paper, we consider the problem of devising blocking schemes for entity matching. There is a lot of work on blocking techniques for supporting various kinds of predicates, e.g. exact matches, fuzzy string-similarity matches, and spatial matches. However, given a complex entity matching function in the form of a Boolean expression over several such predicates, we show that it is an important and non-trivial problem to combine the individual blocking techniques into an efficient blocking scheme for the entity matching function, a problem that has not been studied previously.

User modeling on the Web has rested on the fundamental assumption of Markovian behavior --- a user's next action depends only on her current state, and not the history leading up to the current state. This forms the underpinning of PageRank web ranking, as well as a number of techniques for targeting advertising to users. In this work we examine the validity of this assumption, using data from a number of Web settings. Our main result invokes statistical order estimation tests for Markov chains to establish that Web users are not, in fact, Markovian. We study the extent to which the Markovian assumption is invalid, and derive a number of avenues for further research.

Locality-sensitive hashing (LSH) is a basic primitive in several large-scale data processing applications, including nearest-neighbor search, de-duplication, clustering, etc. In this paper we propose a new and simple method to speed up the widely-used Euclidean realization of LSH. At the heart of our method is a fast way to estimate the Euclidean distance between two d-dimensional vectors; this is achieved by the use of randomized Hadamard transforms in a non-linear setting. This decreases the running time of a (k, L)-parameterized LSH from O(dkL) to O(dlog d + kL). Our experiments show that using the new LSH in nearest-neighbor applications can improve their running times by significant amounts. To the best of our knowledge, this is the first running time improvement to LSH that is both provable and practical.

Entities on social systems, such as users on Twitter, and images on Flickr, are at the core of many interesting applications: they can be ranked in search results, recommended to users, or used in contextual advertising. Such applications assume knowledge of an entity's nature and characteristic attributes. An effective way to encode such knowledge is in the form of tags. An untagged entity is practically inaccessible, since it is hard to retrieve or interact with. To address this, some platforms allow users to manually tag entities. However,while such tags can be informative, they can oftentimes be inadequate, trivial, ambiguous, or even plain false. Numerous automated tagging methods have been proposed to address these issues. However,most of them require pre-existing high-quality tags or descriptive texts for every entity that needs to be tagged. In our work, we propose a method based on social endorsements that is free from such constraints.

The map-reduce paradigm is now standard in industry and academia for processing large-scale data. In this work, we formalize job scheduling in map-reduce as a novel generalization of the two-stage classical flexible flow shop (FFS) problem: instead of a single task at each stage, a job now consists of a set of tasks per stage. For this generalization, we consider the problem of minimizing the total flowtime and give an efficient 12-approximation in the offline setting and an online (1+µ)-speed O(1/µ2)-competitive algorithm.

Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a sparse version of the fundamental tool in dimension reduction -- the Johnson-Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just ~O(1/ε) non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of Ω(1/ε2) both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an ~O(1/ε) update time per non-zero element for a (1 ε)-approximate projection, thereby substantially outperforming the ~O(1/ε2) update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its ~O(d) worst case running time matches the best approach of Ailon and Liberty.

Several results appeared that show significant reduction in time for matrix multiplication, singular value decomposition as well as linear (lscr2) regression, all based on data dependent random sampling. Our key idea is that low dimensional embeddings can be used to eliminate data dependence and provide more versatile, linear time pass efficient matrix computation. Our main contribution is summarized as follows. 1) Independent of the results of Har-Peled and of Deshpande and Vempala, one of the first - and to the best of our knowledge the most efficient - relative error (1 + epsi) parA $AkparFapproximation algorithms for the singular value decomposition of an m times n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O((M(k/epsi+k log k) + (n+m)(k/epsi+k log k)2)log (1/sigma)). 2) The first o(nd2) time (1 + epsi) relative error approximation algorithm for n times d linear (lscr2) regression. 3) A matrix multiplication and norm approximation algorithm that easily applies to implicitly given matrices and can be used as a black box probability boosting tool

Personalized PageRank expresses link-based page quality around user selected pages. The only previous personalized PageRank algorithm that can serve on-line queries for an unrestricted choice of pages on large graphs is our Monte Carlo algorithm [WAW 2004]. In this paper we achieve unrestricted personalization by combining rounding and randomized sketching techniques in the dynamic programming algorithm of Jeh and Widom [WWW 2003]. We evaluate the precision of approximation experimentally on large scale real-world data and find significant improvement over previous results. As a key theoretical contribution we show that our algorithms use an optimal amount of space by also improving earlier asymptotic worst-case lower bounds. Our lower bounds and algorithms apply to the SimRank as well; of independent interest is the reduction of the SimRank computation to personalized PageRank.