Alex Gittens

Stephen Tu, Shivaram Venkataraman, Ashia C. Wilson et al.

Recent work by Nesterov and Stich showed that momentum can be used to accelerate the rate of convergence for block Gauss-Seidel in the setting where a fixed partitioning of the coordinates is chosen ahead of time. We show that this setting is too restrictive, constructing instances where breaking locality by running non-accelerated Gauss-Seidel with randomly sampled coordinates substantially outperforms accelerated Gauss-Seidel with any fixed partitioning. Motivated by this finding, we analyze the accelerated block Gauss-Seidel algorithm in the random coordinate sampling setting. Our analysis captures the benefit of acceleration with a new data-dependent parameter which is well behaved when the matrix sub-blocks are well-conditioned. Empirically, we show that accelerated Gauss-Seidel with random coordinate sampling provides speedups for large scale machine learning tasks when compared to non-accelerated Gauss-Seidel and the classical conjugate-gradient algorithm.

Anik Saha, Oktie Hassanzadeh, Alex Gittens et al. · ibm-research

Neural ranking methods based on large transformer models have recently gained significant attention in the information retrieval community, and have been adopted by major commercial solutions. Nevertheless, they are computationally expensive to create, and require a great deal of labeled data for specialized corpora. In this paper, we explore a low resource alternative which is a bag-of-embedding model for document retrieval and find that it is competitive with large transformer models fine tuned on information retrieval tasks. Our results show that a simple combination of TF-IDF, a traditional keyword matching method, with a shallow embedding model provides a low cost path to compete well with the performance of complex neural ranking models on 3 datasets. Furthermore, adding TF-IDF measures improves the performance of large-scale fine tuned models on these tasks.

Christos Boutsidis, Alex Gittens

Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the Subsampled Randomized Hadamard Transform (SRHT). In this article, we address the efficacy, in the Frobenius and spectral norms, of an SRHT-based low-rank matrix approximation technique introduced by Woolfe, Liberty, Rohklin, and Tygert. We establish a slightly better Frobenius norm error bound than currently available, and a much sharper spectral norm error bound (in the presence of reasonable decay of the singular values). Along the way, we produce several results on matrix operations with SRHTs (such as approximate matrix multiplication) that may be of independent interest. Our approach builds upon Tropp's in "Improved analysis of the Subsampled Randomized Hadamard Transform".

Anik Saha, Oktie Hassanzadeh, Alex Gittens et al. · ibm-research

Causal knowledge extraction is the task of extracting relevant causes and effects from text by detecting the causal relation. Although this task is important for language understanding and knowledge discovery, recent works in this domain have largely focused on binary classification of a text segment as causal or non-causal. In this regard, we perform a thorough analysis of three sequence tagging models for causal knowledge extraction and compare it with a span based approach to causality extraction. Our experiments show that embeddings from pre-trained language models (e.g. BERT) provide a significant performance boost on this task compared to previous state-of-the-art models with complex architectures. We observe that span based models perform better than simple sequence tagging models based on BERT across all 4 data sets from diverse domains with different types of cause-effect phrases.

Lilian Ngweta, Subha Maity, Alex Gittens et al.

Learning visual representations with interpretable features, i.e., disentangled representations, remains a challenging problem. Existing methods demonstrate some success but are hard to apply to large-scale vision datasets like ImageNet. In this work, we propose a simple post-processing framework to disentangle content and style in learned representations from pre-trained vision models. We model the pre-trained features probabilistically as linearly entangled combinations of the latent content and style factors and develop a simple disentanglement algorithm based on the probabilistic model. We show that the method provably disentangles content and style features and verify its efficacy empirically. Our post-processed features yield significant domain generalization performance improvements when the distribution shift occurs due to style changes or style-related spurious correlations.

Huzaifa Arif, Pin-Yu Chen, Alex Gittens et al.

With the increasing reliance on AI models for weather forecasting, it is imperative to evaluate their vulnerability to adversarial perturbations. This work introduces Weather Adaptive Adversarial Perturbation Optimization (WAAPO), a novel framework for generating targeted adversarial perturbations that are both effective in manipulating forecasts and stealthy to avoid detection. WAAPO achieves this by incorporating constraints for channel sparsity, spatial localization, and smoothness, ensuring that perturbations remain physically realistic and imperceptible. Using the ERA5 dataset and FourCastNet (Pathak et al. 2022), we demonstrate WAAPO's ability to generate adversarial trajectories that align closely with predefined targets, even under constrained conditions. Our experiments highlight critical vulnerabilities in AI-driven forecasting models, where small perturbations to initial conditions can result in significant deviations in predicted weather patterns. These findings underscore the need for robust safeguards to protect against adversarial exploitation in operational forecasting systems.

Anik Saha, Alex Gittens, Bulent Yener

Pre-trained contextual language models are ubiquitously employed for language understanding tasks, but are unsuitable for resource-constrained systems. Noncontextual word embeddings are an efficient alternative in these settings. Such methods typically use one vector to encode multiple different meanings of a word, and incur errors due to polysemy. This paper proposes a two-stage method to distill multiple word senses from a pre-trained language model (BERT) by using attention over the senses of a word in a context and transferring this sense information to fit multi-sense embeddings in a skip-gram-like framework. We demonstrate an effective approach to training the sense disambiguation mechanism in our model with a distribution over word senses extracted from the output layer embeddings of BERT. Experiments on the contextual word similarity and sense induction tasks show that this method is superior to or competitive with state-of-the-art multi-sense embeddings on multiple benchmark data sets, and experiments with an embedding-based topic model (ETM) demonstrates the benefits of using this multi-sense embedding in a downstream application.

Deniz Koyuncu, Alex Gittens, Bülent Yener et al.

Adversarial Missingness (AM) attacks aim to manipulate model fitting by carefully engineering a missing data problem to achieve a specific malicious objective. AM attacks are significantly different from prior data poisoning attacks in that no malicious data inserted and no data is maliciously perturbed. Current AM attacks are feasible only under the assumption that the modeler (victim) uses full-information maximum likelihood methods to handle missingness. This work aims to remedy this limitation of AM attacks; in the approach taken here, the adversary achieves their goal by solving a bi-level optimization problem to engineer the adversarial missingness mechanism, where the lower level problem incorporates a differentiable approximation of the targeted missingness remediation technique. As instantiations of this framework, AM attacks are provided for three popular techniques: (i) complete case analysis, (ii) mean imputation, and (iii) regression-based imputation for general empirical risk minimization (ERM) problems. Experiments on real-world data show that AM attacks are successful with modest levels of missingness (less than 20%). Furthermore, we show on the real-world Twins dataset that AM attacks can manipulate the estimated average treatment effect (ATE) as an instance of the general ERM problems: the adversary succeeds in not only reversing the sign, but also in substantially inflating the ATE values from a true value of -1.61% to a manipulated one as high as 10%. These experimental results hold when the ATE is calculated using multiple regression-based estimators with different architectures, even when the adversary is restricted to modifying only a subset of the training data.

Arvind Rathnashyam, Alex Gittens

We derive approximation bounds for learning single neuron models using thresholded gradient descent when both the labels and the covariates are possibly corrupted adversarially. We assume the data follows the model $y = σ(\mathbf{w}^{*} \cdot \mathbf{x}) + ξ,$ where $σ$ is a nonlinear activation function, the noise $ξ$ is Gaussian, and the covariate vector $\mathbf{x}$ is sampled from a sub-Gaussian distribution. We study sigmoidal, leaky-ReLU, and ReLU activation functions and derive a $O(ν\sqrt{ε\log(1/ε)})$ approximation bound in $\ell_{2}$-norm, with sample complexity $O(d/ε)$ and failure probability $e^{-Ω(d)}$. We also study the linear regression problem, where $σ(\mathbf{x}) = \mathbf{x}$. We derive a $O(νε\log(1/ε))$ approximation bound, improving upon the previous $O(ν)$ approximation bounds for the gradient-descent based iterative thresholding algorithms of Bhatia et al. (NeurIPS 2015) and Shen and Sanghavi (ICML 2019). Our algorithm has a $O(\textrm{polylog}(N,d)\log(R/ε))$ runtime complexity when $\|\mathbf{w}^{*}\|_2 \leq R$, improving upon the $O(\text{polylog}(N,d)/ε^2)$ runtime complexity of Awasthi et al. (NeurIPS 2022).

Huzaifa Arif, Keerthiram Murugesan, Ching-Yun Ko et al.

We propose patching for large language models (LLMs) like software versions, a lightweight and modular approach for addressing safety vulnerabilities. While vendors release improved LLM versions, major releases are costly, infrequent, and difficult to tailor to customer needs, leaving released models with known safety gaps. Unlike full-model fine-tuning or major version updates, our method enables rapid remediation by prepending a compact, learnable prefix to an existing model. This "patch" introduces only 0.003% additional parameters, yet reliably steers model behavior toward that of a safer reference model. Across three critical domains (toxicity mitigation, bias reduction, and harmfulness refusal) policy patches achieve safety improvements comparable to next-generation safety-aligned models while preserving fluency. Our results demonstrate that LLMs can be "patched" much like software, offering vendors and practitioners a practical mechanism for distributing scalable, efficient, and composable safety updates between major model releases.

Huzaifa Arif, Keerthiram Murugesan, Payel Das et al.

This paper explores inference-time data leakage risks of deep neural networks (NNs), where a curious and honest model service provider is interested in retrieving users' private data inputs solely based on the model inference results. Particularly, we revisit residual NNs due to their popularity in computer vision and our hypothesis that residual blocks are a primary cause of data leakage owing to the use of skip connections. By formulating inference-time data leakage as a constrained optimization problem, we propose a novel backward feature inversion method, \textbf{PEEL}, which can effectively recover block-wise input features from the intermediate output of residual NNs. The surprising results in high-quality input data recovery can be explained by the intuition that the output from these residual blocks can be considered as a noisy version of the input and thus the output retains sufficient information for input recovery. We demonstrate the effectiveness of our layer-by-layer feature inversion method on facial image datasets and pre-trained classifiers. Our results show that PEEL outperforms the state-of-the-art recovery methods by an order of magnitude when evaluated by mean squared error (MSE). The code is available at \href{https://github.com/Huzaifa-Arif/PEEL}{https://github.com/Huzaifa-Arif/PEEL}

Gabriel Orlanski, Alex Gittens

Answering a programming question using only its title is difficult as salient contextual information is omitted. Based on this observation, we present a corpus of over 40,000 StackOverflow question texts to be used in conjunction with their corresponding intents from the CoNaLa dataset (Yin et al., 2018). Using both the intent and question body, we use BART to establish a baseline BLEU score of 34.35 for this new task. We find further improvements of $2.8\%$ by combining the mined CoNaLa data with the labeled data to achieve a 35.32 BLEU score. We evaluate prior state-of-the-art CoNaLa models with this additional data and find that our proposed method of using the body and mined data beats the BLEU score of the prior state-of-the-art by $71.96\%$. Finally, we perform ablations to demonstrate that BART is an unsupervised multimodal learner and examine its extractive behavior. The code and data can be found https://github.com/gabeorlanski/stackoverflow-encourages-cheating.

Nidhi Rastogi, Sharmishtha Dutta, Mohammed J. Zaki et al.

Threat intelligence on malware attacks and campaigns is increasingly being shared with other security experts for a cost or for free. Other security analysts use this intelligence to inform them of indicators of compromise, attack techniques, and preventative actions. Security analysts prepare threat analysis reports after investigating an attack, an emerging cyber threat, or a recently discovered vulnerability. Collectively known as cyber threat intelligence (CTI), the reports are typically in an unstructured format and, therefore, challenging to integrate seamlessly into existing intrusion detection systems. This paper proposes a framework that uses the aggregated CTI for analysis and defense at scale. The information is extracted and stored in a structured format using knowledge graphs such that the semantics of the threat intelligence can be preserved and shared at scale with other security analysts. Specifically, we propose the first semi-supervised open-source knowledge graph-based framework, TINKER, to capture cyber threat information and its context. Following TINKER, we generate a Cyberthreat Intelligence Knowledge Graph (CTI-KG) and demonstrate the usage using different use cases.

Nidhi Rastogi, Sharmishtha Dutta, Mohammed J. Zaki et al.

Malware threat intelligence uncovers deep information about malware, threat actors, and their tactics, Indicators of Compromise(IoC), and vulnerabilities in different platforms from scattered threat sources. This collective information can guide decision making in cyber defense applications utilized by security operation centers(SoCs). In this paper, we introduce an open-source malware ontology - MALOnt that allows the structured extraction of information and knowledge graph generation, especially for threat intelligence. The knowledge graph that uses MALOnt is instantiated from a corpus comprising hundreds of annotated malware threat reports. The knowledge graph enables the analysis, detection, classification, and attribution of cyber threats caused by malware. We also demonstrate the annotation process using MALOnt on exemplar threat intelligence reports. A work in progress, this research is part of a larger effort towards auto-generation of knowledge graphs (KGs)for gathering malware threat intelligence from heterogeneous online resources.

Lilian Ngweta, Mayank Agarwal, Subha Maity et al.

Large Language Models (LLMs) need to be aligned with human expectations to ensure their safety and utility in most applications. Alignment is challenging, costly, and needs to be repeated for every LLM and alignment criterion. We propose to decouple LLMs and alignment by training aligner models that can be used to align any LLM for a given criteria on an as-needed basis, thus also reducing the potential negative impacts of alignment on performance. Our recipe for training the aligner models solely relies on synthetic data generated with a (prompted) LLM and can be easily adjusted for a variety of alignment criteria. We use the same synthetic data to train inspectors, binary miss-alignment classification models to guide a "squad" of multiple aligners. Our empirical results demonstrate consistent improvements when applying aligner squad to various LLMs, including chat-aligned models, across several instruction-following and red-teaming datasets.

Deniz Koyuncu, Alex Gittens, Bülent Yener et al.

Inference of causal structures from observational data is a key component of causal machine learning; in practice, this data may be incompletely observed. Prior work has demonstrated that adversarial perturbations of completely observed training data may be used to force the learning of inaccurate causal structural models (SCMs). However, when the data can be audited for correctness (e.g., it is crytographically signed by its source), this adversarial mechanism is invalidated. This work introduces a novel attack methodology wherein the adversary deceptively omits a portion of the true training data to bias the learned causal structures in a desired manner. Theoretically sound attack mechanisms are derived for the case of arbitrary SCMs, and a sample-efficient learning-based heuristic is given for Gaussian SCMs. Experimental validation of these approaches on real and synthetic data sets demonstrates the effectiveness of adversarial missingness attacks at deceiving popular causal structure learning algorithms.

Alex Gittens, Malik Magdon-Ismail

Given data ${\rm X}\in\mathbb{R}^{n\times d}$ and labels $\mathbf{y}\in\mathbb{R}^{n}$ the goal is find $\mathbf{w}\in\mathbb{R}^d$ to minimize $\Vert{\rm X}\mathbf{w}-\mathbf{y}\Vert^2$. We give a polynomial algorithm that, \emph{oblivious to $\mathbf{y}$}, throws out $n/(d+\sqrt{n})$ data points and is a $(1+d/n)$-approximation to optimal in expectation. The motivation is tight approximation with reduced label complexity (number of labels revealed). We reduce label complexity by $Ω(\sqrt{n})$. Open question: Can label complexity be reduced by $Ω(n)$ with tight $(1+d/n)$-approximation?

Deniz Koyuncu, Alex Gittens, Bülent Yener

One limitation of the most statistical/machine learning-based variable selection approaches is their inability to control the false selections. A recently introduced framework, model-x knockoffs, provides that to a wide range of models but lacks support for datasets with missing values. In this work, we discuss ways of preserving the theoretical guarantees of the model-x framework in the missing data setting. First, we prove that posterior sampled imputation allows reusing existing knockoff samplers in the presence of missing values. Second, we show that sampling knockoffs only for the observed variables and applying univariate imputation also preserves the false selection guarantees. Third, for the special case of latent variable models, we demonstrate how jointly imputing and sampling knockoffs can reduce the computational complexity. We have verified the theoretical findings with two different exploratory variable distributions and investigated how the missing data pattern, amount of correlation, the number of observations, and missing values affected the statistical power.

Daniel Park, Haidar Khan, Azer Khan et al.

Adversarial examples pose a threat to deep neural network models in a variety of scenarios, from settings where the adversary has complete knowledge of the model in a "white box" setting and to the opposite in a "black box" setting. In this paper, we explore the use of output randomization as a defense against attacks in both the black box and white box models and propose two defenses. In the first defense, we propose output randomization at test time to thwart finite difference attacks in black box settings. Since this type of attack relies on repeated queries to the model to estimate gradients, we investigate the use of randomization to thwart such adversaries from successfully creating adversarial examples. We empirically show that this defense can limit the success rate of a black box adversary using the Zeroth Order Optimization attack to 0%. Secondly, we propose output randomization training as a defense against white box adversaries. Unlike prior approaches that use randomization, our defense does not require its use at test time, eliminating the Backward Pass Differentiable Approximation attack, which was shown to be effective against other randomization defenses. Additionally, this defense has low overhead and is easily implemented, allowing it to be used together with other defenses across various model architectures. We evaluate output randomization training against the Projected Gradient Descent attacker and show that the defense can reduce the PGD attack's success rate down to 12% when using cross-entropy loss.

Kevin Kim, Alex Gittens

This work proposes to learn fair low-rank tensor decompositions by regularizing the Canonical Polyadic Decomposition factorization with the kernel Hilbert-Schmidt independence criterion (KHSIC). It is shown, theoretically and empirically, that a small KHSIC between a latent factor and the sensitive features guarantees approximate statistical parity. The proposed algorithm surpasses the state-of-the-art algorithm, FATR (Zhu et al., 2018), in controlling the trade-off between fairness and residual fit on synthetic and real data sets.

Dong Hu, Alex Gittens, Malik Magdon-Ismail

Matrix completion is a ubiquitous tool in machine learning and data analysis. Most work in this area has focused on the number of observations necessary to obtain an accurate low-rank approximation. In practice, however, the cost of observations is an important limiting factor, and experimentalists may have on hand multiple modes of observation with differing noise-vs-cost trade-offs. This paper considers matrix completion subject to such constraints: a budget is imposed and the experimentalist's goal is to allocate this budget between two sampling modalities in order to recover an accurate low-rank approximation. Specifically, we consider that it is possible to obtain low noise, high cost observations of individual entries or high noise, low cost observations of entire columns. We introduce a regression-based completion algorithm for this setting and experimentally verify the performance of our approach on both synthetic and real data sets. When the budget is low, our algorithm outperforms standard completion algorithms. When the budget is high, our algorithm has comparable error to standard nuclear norm completion algorithms and requires much less computational effort.

Malik Magdon-Ismail, Alex Gittens

We give a fast oblivious L2-embedding of $A\in \mathbb{R}^{n x d}$ to $B\in \mathbb{R}^{r x d}$ satisfying $(1-\varepsilon)\|A x\|_2^2 \le \|B x\|_2^2 <= (1+\varepsilon) \|Ax\|_2^2.$ Our embedding dimension $r$ equals $d$, a constant independent of the distortion $\varepsilon$. We use as a black-box any L2-embedding $Π^T A$ and inherit its runtime and accuracy, effectively decoupling the dimension $r$ from runtime and accuracy, allowing downstream machine learning applications to benefit from both a low dimension and high accuracy (in prior embeddings higher accuracy means higher dimension). We give applications of our L2-embedding to regression, PCA and statistical leverage scores. We also give applications to L1: 1.) An oblivious L1-embedding with dimension $d+O(d\ln^{1+η} d)$ and distortion $O((d\ln d)/\ln\ln d)$, with application to constructing well-conditioned bases; 2.) Fast approximation of L1-Lewis weights using our L2 embedding to quickly approximate L2-leverage scores.

Shusen Wang, Alex Gittens, Michael W. Mahoney

Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel approximation, e.g., the Nyström method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel $k$-means clustering, and shows that applying the linear $k$-means clustering algorithm to $\frac{k}ε (1 + o(1))$ features constructed using a so-called rank-restricted Nyström approximation results in cluster assignments that satisfy a $1 + ε$ approximation ratio in terms of the kernel $k$-means cost function, relative to the guarantee provided by the same algorithm without the use of the Nyström method. As part of the analysis, this work establishes a novel $1 + ε$ relative-error trace norm guarantee for low-rank approximation using the rank-restricted Nyström approximation. Empirical evaluations on the $8.1$ million instance MNIST8M dataset demonstrate the scalability and usefulness of kernel $k$-means clustering with Nyström approximation. This work argues that spectral clustering using Nyström approximation---a popular and computationally efficient, but theoretically unsound approach to non-linear clustering---should be replaced with the efficient and theoretically sound combination of kernel $k$-means clustering with Nyström approximation. The superior performance of the latter approach is empirically verified.

Shusen Wang, Alex Gittens, Michael W. Mahoney

We address the statistical and optimization impacts of the classical sketch and Hessian sketch used to approximately solve the Matrix Ridge Regression (MRR) problem. Prior research has quantified the effects of classical sketch on the strictly simpler least squares regression (LSR) problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR: namely, it recovers nearly optimal solutions. By contrast, Hessian sketch does not have this guarantee, instead, the approximation error is governed by a subtle interplay between the "mass" in the responses and the optimal objective value. For both types of approximation, the regularization in the sketched MRR problem results in significantly different statistical properties from those of the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the bias and variance of sketched MRR, these bounds show that classical sketch significantly increases the variance, while Hessian sketch significantly increases the bias. Empirically, sketched MRR solutions can have risks that are higher by an order-of-magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases the gap between the risks of the true and sketched solutions to the MRR problem. Thus, in parallel or distributed settings, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the increased statistical risk incurred by sketching.

Jiyan Yang, Alex Gittens

Recent years have demonstrated that using random feature maps can significantly decrease the training and testing times of kernel-based algorithms without significantly lowering their accuracy. Regrettably, because random features are target-agnostic, typically thousands of such features are necessary to achieve acceptable accuracies. In this work, we consider the problem of learning a small number of explicit polynomial features. Our approach, named Tensor Machines, finds a parsimonious set of features by optimizing over the hypothesis class introduced by Kar and Karnick for random feature maps in a target-specific manner. Exploiting a natural connection between polynomials and tensors, we provide bounds on the generalization error of Tensor Machines. Empirically, Tensor Machines behave favorably on several real-world datasets compared to other state-of-the-art techniques for learning polynomial features, and deliver significantly more parsimonious models.

Da Kuang, Alex Gittens, Raffay Hamid

The dominant cost in solving least-square problems using Newton's method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter ($λ$). We propose an efficient way to interpolate the Cholesky factors of the Hessian matrix computed over a small set of $λ$ values. This approximation enables us to optimally minimize the hold-out error while incurring only a fraction of the cost compared to exact cross-validation. We provide a formal error bound for our approximation scheme and present solutions to a set of key implementation challenges that allow our approach to maximally exploit the compute power of modern architectures. We present a thorough empirical analysis over multiple datasets to show the effectiveness of our approach.

Raffay Hamid, Ying Xiao, Alex Gittens et al.

Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not utilize the capacity of the projected feature space effectively. To address this challenge, we propose compact random feature maps (CRAFTMaps) to approximate polynomial kernels more concisely and accurately. We prove the error bounds of CRAFTMaps demonstrating their superior kernel reconstruction performance compared to the previous approximation schemes. We show how structured random matrices can be used to efficiently generate CRAFTMaps, and present a single-pass algorithm using CRAFTMaps to learn non-linear multi-class classifiers. We present experiments on multiple standard data-sets with performance competitive with state-of-the-art results.

Christos Boutsidis, Alex Gittens, Prabhanjan Kambadur

Spectral clustering is one of the most important algorithms in data mining and machine intelligence; however, its computational complexity limits its application to truly large scale data analysis. The computational bottleneck in spectral clustering is computing a few of the top eigenvectors of the (normalized) Laplacian matrix corresponding to the graph representing the data to be clustered. One way to speed up the computation of these eigenvectors is to use the "power method" from the numerical linear algebra literature. Although the power method has been empirically used to speed up spectral clustering, the theory behind this approach, to the best of our knowledge, remains unexplored. This paper provides the \emph{first} such rigorous theoretical justification, arguing that a small number of power iterations suffices to obtain near-optimal partitionings using the approximate eigenvectors. Specifically, we prove that solving the $k$-means clustering problem on the approximate eigenvectors obtained via the power method gives an additive-error approximation to solving the $k$-means problem on the optimal eigenvectors.

Alex Gittens, Michael W. Mahoney

We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds---e.g. improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error---and they point to future directions to make these algorithms useful in even larger-scale machine learning applications.

Alex Gittens, Joel A. Tropp

Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the focus has been--with few exceptions--on ensuring the quality of approximation in the spectral and Frobenius norms. For certain graph algorithms, however, the $(\infty,1)$ norm may be a more natural measure of performance. This paper addresses the problem of approximating a real matrix A by a sparse random matrix X with respect to several norms. It provides the first results on approximation error in the $(\infty, 1)$ and $(\infty, 2)$ norms, and it uses a result of Latala to study approximation error in the spectral norm. These bounds hold for random sparsification schemes which ensure that the entries of X are independent and average to the corresponding entries of A. Optimality of the $(\infty, 1)$ and $(\infty,2)$ error estimates is established. Concentration results for the three norms hold when the entries of X are uniformly bounded. The spectral error bound is used to predict the performance of several sparsification and quantization schemes that have appeared in the literature; the results are competitive with the performance guarantees given by earlier scheme-specific analyses.