Yan Shuo Tan

Abhineet Agarwal, Ana M. Kenney, Yan Shuo Tan et al. · berkeley

Random forests (RFs) are among the most popular supervised learning algorithms due to their nonlinear flexibility and ease-of-use. However, as black box models, they can only be interpreted via algorithmically-defined feature importance methods, such as Mean Decrease in Impurity (MDI), which have been observed to be highly unstable and have ambiguous scientific meaning. Furthermore, they can perform poorly in the presence of smooth or additive structure. To address this, we reinterpret decision trees and MDI as linear regression and $R^2$ values, respectively, with respect to engineered features associated with the tree's decision splits. This allows us to combine the respective strengths of RFs and generalized linear models in a framework called RF+, which also yields an improved feature importance method we call MDI+. Through extensive data-inspired simulations and real-world datasets, we show that RF+ improves prediction accuracy over RFs and that MDI+ outperforms popular feature importance measures in identifying signal features, often yielding more than a 10% improvement over its closest competitor. In case studies on drug response prediction and breast cancer subtyping, we further show that MDI+ extracts well-established genes with significantly greater stability compared to existing feature importance measures.

Yan Shuo Tan, Kenyon Ng, Ruizhe Deng et al.

Thompson sampling is a widely used strategy for contextual bandits: at each round, it samples a reward function from a Bayesian posterior and acts greedily under that sample. Prior-data fitted networks (PFNs), such as TabPFN v2+ and TabICL v2, are attractive candidates for this purpose because they approximate Bayesian posterior predictive distributions in a single forward pass. However, PFNs predict noisy future rewards, while Thompson sampling requires uncertainty over the latent mean reward function. We propose PFN-TS, a Thompson sampling algorithm that converts PFN posterior predictives into mean-reward samples using a subsampled predictive central limit theorem. The method estimates posterior variance from a geometric grid of $O(\log n)$ dataset prefixes rather than the full $O(n)$ predictive sequence used in previous predictive-sequence approaches, and reuses TabICL's cached representations across rounds. We prove consistency of the subsampled variance estimator and give a Bayesian regret bound that decomposes PFN-TS regret into exact posterior-sampling regret under the PFN prior plus approximation terms. Empirically, PFN-TS achieves the best average rank across nonlinear synthetic and OpenML classification-to-bandit benchmarks, remains competitive on linear and BART-generated rewards, and attains the highest estimated policy value in an offline mobile-health evaluation. Code is available at https://anonymous.4open.science/r/PFN_TS-36ED/.

Tianqi Zhao, Guanyang Wang, Yan Shuo Tan et al.

Clustering tabular data is a fundamental yet challenging problem due to heterogeneous feature types, diverse data-generating mechanisms, and the absence of transferable inductive biases across datasets. Prior-fitted networks (PFNs) have recently demonstrated strong generalization in supervised tabular learning by amortizing Bayesian inference under a broad synthetic prior. Extending this paradigm to clustering is nontrivial: clustering is unsupervised, admits a combinatorial and permutation-invariant output space, and requires inferring the number of clusters. We introduce TabClustPFN, a prior-fitted network for tabular data clustering that performs amortized Bayesian inference over both cluster assignments and cluster cardinality. Pretrained on synthetic datasets drawn from a flexible clustering prior, TabClustPFN clusters unseen datasets in a single forward pass, without dataset-specific retraining or hyperparameter tuning. The model naturally handles heterogeneous numerical and categorical features and adapts to a wide range of clustering structures. Experiments on synthetic data and curated real-world tabular benchmarks show that TabClustPFN outperforms classical, deep, and amortized clustering baselines, while exhibiting strong robustness in out-of-the-box exploratory settings. Code is available at https://github.com/Tianqi-Zhao/TabClustPFN.

Omer Ronen, Theo Saarinen, Yan Shuo Tan et al.

Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression algorithm. The posterior is a distribution over sums of decision trees, and predictions are made by averaging approximate samples from the posterior. The combination of strong predictive performance and the ability to provide uncertainty measures has led BART to be commonly used in the social sciences, biostatistics, and causal inference. BART uses Markov Chain Monte Carlo (MCMC) to obtain approximate posterior samples over a parameterized space of sums of trees, but it has often been observed that the chains are slow to mix. In this paper, we provide the first lower bound on the mixing time for a simplified version of BART in which we reduce the sum to a single tree and use a subset of the possible moves for the MCMC proposal distribution. Our lower bound for the mixing time grows exponentially with the number of data points. Inspired by this new connection between the mixing time and the number of data points, we perform rigorous simulations on BART. We show qualitatively that BART's mixing time increases with the number of data points. The slow mixing time of the simplified BART suggests a large variation between different runs of the simplified BART algorithm and a similar large variation is known for BART in the literature. This large variation could result in a lack of stability in the models, predictions, and posterior intervals obtained from the BART MCMC samples. Our lower bound and simulations suggest increasing the number of chains with the number of data points.

Xin Chen, Jason M. Klusowski, Yan Shuo Tan

Stacking regressions is an ensemble technique that forms linear combinations of different regression estimators to enhance predictive accuracy. The conventional approach uses cross-validation data to generate predictions from the constituent estimators, and least-squares with nonnegativity constraints to learn the combination weights. In this paper, we learn these weights analogously by minimizing a regularized version of the empirical risk subject to a nonnegativity constraint. When the constituent estimators are linear least-squares projections onto nested subspaces separated by at least three dimensions, we show that thanks to an adaptive shrinkage effect, the resulting stacked estimator has strictly smaller population risk than best single estimator among them, with more significant gains when the signal-to-noise ratio is small. Here "best" refers to an estimator that minimizes a model selection criterion such as AIC or BIC. In other words, in this setting, the best single estimator is inadmissible. Because the optimization problem can be reformulated as isotonic regression, the stacked estimator requires the same order of computation as the best single estimator, making it an attractive alternative in terms of both performance and implementation.

Jean Feng, Avni Kothari, Patrick Vossler et al.

Developing safe, effective, and practically useful clinical prediction models (CPMs) traditionally requires iterative collaboration between clinical experts, data scientists, and informaticists. This process refines the often small but critical details of the model building process, such as which features/patients to include and how clinical categories should be defined. However, this traditional collaboration process is extremely time- and resource-intensive, resulting in only a small fraction of CPMs reaching clinical practice. This challenge intensifies when teams attempt to incorporate unstructured clinical notes, which can contain an enormous number of concepts. To address this challenge, we introduce HACHI, an iterative human-in-the-loop framework that uses AI agents to accelerate the development of fully interpretable CPMs by enabling the exploration of concepts in clinical notes. HACHI alternates between (i) an AI agent rapidly exploring and evaluating candidate concepts in clinical notes and (ii) clinical and domain experts providing feedback to improve the CPM learning process. HACHI defines concepts as simple yes-no questions that are used in linear models, allowing the clinical AI team to transparently review, refine, and validate the CPM learned in each round. In two real-world prediction tasks (acute kidney injury and traumatic brain injury), HACHI outperforms existing approaches, surfaces new clinically relevant concepts not included in commonly-used CPMs, and improves model generalizability across clinical sites and time periods. Furthermore, HACHI reveals the critical role of the clinical AI team, such as directing the AI agent to explore concepts that it had not previously considered, adjusting the granularity of concepts it considers, changing the objective function to better align with the clinical objectives, and identifying issues of data bias and leakage.

Abhineet Agarwal, Yan Shuo Tan, Omer Ronen et al.

Tree-based models such as decision trees and random forests (RF) are a cornerstone of modern machine-learning practice. To mitigate overfitting, trees are typically regularized by a variety of techniques that modify their structure (e.g. pruning). We introduce Hierarchical Shrinkage (HS), a post-hoc algorithm that does not modify the tree structure, and instead regularizes the tree by shrinking the prediction over each node towards the sample means of its ancestors. The amount of shrinkage is controlled by a single regularization parameter and the number of data points in each ancestor. Since HS is a post-hoc method, it is extremely fast, compatible with any tree growing algorithm, and can be used synergistically with other regularization techniques. Extensive experiments over a wide variety of real-world datasets show that HS substantially increases the predictive performance of decision trees, even when used in conjunction with other regularization techniques. Moreover, we find that applying HS to each tree in an RF often improves accuracy, as well as its interpretability by simplifying and stabilizing its decision boundaries and SHAP values. We further explain the success of HS in improving prediction performance by showing its equivalence to ridge regression on a (supervised) basis constructed of decision stumps associated with the internal nodes of a tree. All code and models are released in a full-fledged package available on Github (github.com/csinva/imodels)

Chandan Singh, Yan Shuo Tan, Weijia Xu et al.

Agentic data science (ADS) systems are rapidly improving their capability to autonomously analyze, fit, and interpret data, potentially moving towards a future where agents conduct the vast majority of data-science work. However, current ADS systems use statistical tools designed to be interpretable by humans, rather than interpretable by agents. To address this, we introduce Agentic-imodels, an agentic autoresearch loop that evolves data-science tools designed to be interpretable by agents. Specifically, it develops a library of scikit-learn-compatible regressors for tabular data that are optimized for both predictive performance and a novel LLM-based interpretability metric. The metric measures a suite of LLM-graded tests that probe whether a fitted model's string representation is "simulatable" by an LLM, i.e. whether the LLM can answer questions about the model's behavior by reading its string output alone. We find that the evolved models jointly improve predictive performance and agent-facing interpretability, generalizing to new datasets and new interpretability tests. Furthermore, these evolved models improve downstream end-to-end ADS, increasing performance for Copilot CLI, Claude Code, and Codex on the BLADE benchmark by up to 73%

Qiong Zhang, Yan Shuo Tan, Jiahua Chen

Traditional statistical methods need to be updated to work with modern distributed data storage paradigms. A common approach is the split-and-conquer framework, which involves learning models on local machines and averaging their parameter estimates. However, this does not work for the important problem of learning finite mixture models, because subpopulation indices on each local machine may be arbitrarily permuted (the "label switching problem"). Zhang and Chen (2022) proposed Mixture Reduction (MR) to address this issue, but MR remains vulnerable to Byzantine failure, whereby a fraction of local machines may transmit arbitrarily erroneous information. This paper introduces Distance Filtered Mixture Reduction (DFMR), a Byzantine tolerant adaptation of MR that is both computationally efficient and statistically sound. DFMR leverages the densities of local estimates to construct a robust filtering mechanism. By analysing the pairwise L2 distances between local estimates, DFMR identifies and removes severely corrupted local estimates while retaining the majority of uncorrupted ones. We provide theoretical justification for DFMR, proving its optimal convergence rate and asymptotic equivalence to the global maximum likelihood estimate under standard assumptions. Numerical experiments on simulated and real-world data validate the effectiveness of DFMR in achieving robust and accurate aggregation in the presence of Byzantine failure.

Jean Feng, Avni Kothari, Luke Zier et al.

Concept Bottleneck Models (CBMs) have been proposed as a compromise between white-box and black-box models, aiming to achieve interpretability without sacrificing accuracy. The standard training procedure for CBMs is to predefine a candidate set of human-interpretable concepts, extract their values from the training data, and identify a sparse subset as inputs to a transparent prediction model. However, such approaches are often hampered by the tradeoff between exploring a sufficiently large set of concepts versus controlling the cost of obtaining concept extractions, resulting in a large interpretability-accuracy tradeoff. This work investigates a novel approach that sidesteps these challenges: BC-LLM iteratively searches over a potentially infinite set of concepts within a Bayesian framework, in which Large Language Models (LLMs) serve as both a concept extraction mechanism and prior. Even though LLMs can be miscalibrated and hallucinate, we prove that BC-LLM can provide rigorous statistical inference and uncertainty quantification. Across image, text, and tabular datasets, BC-LLM outperforms interpretable baselines and even black-box models in certain settings, converges more rapidly towards relevant concepts, and is more robust to out-of-distribution samples.

Yan Shuo Tan, Jason M. Klusowski, Krishnakumar Balasubramanian

Models based on recursive adaptive partitioning such as decision trees and their ensembles are popular for high-dimensional regression as they can potentially avoid the curse of dimensionality. Because empirical risk minimization (ERM) is computationally infeasible, these models are typically trained using greedy algorithms. Although effective in many cases, these algorithms have been empirically observed to get stuck at local optima. We explore this phenomenon in the context of learning sparse regression functions over $d$ binary features, showing that when the true regression function $f^*$ does not satisfy Abbe et al. (2022)'s Merged Staircase Property (MSP), greedy training requires $\exp(Ω(d))$ to achieve low estimation error. Conversely, when $f^*$ does satisfy MSP, greedy training can attain small estimation error with only $O(\log d)$ samples. This dichotomy mirrors that of two-layer neural networks trained with stochastic gradient descent (SGD) in the mean-field regime, thereby establishing a head-to-head comparison between SGD-trained neural networks and greedy recursive partitioning estimators. Furthermore, ERM-trained recursive partitioning estimators achieve low estimation error with $O(\log d)$ samples irrespective of whether $f^*$ satisfies MSP, thereby demonstrating a statistical-computational trade-off for greedy training. Our proofs are based on a novel interpretation of greedy recursive partitioning using stochastic process theory and a coupling technique that may be of independent interest.

Zineng Xu, Subhroshekhar Ghosh, Yan Shuo Tan

While globally optimal empirical risk minimization (ERM) decision trees have become computationally feasible and empirically successful, rigorous theoretical guarantees for their statistical performance remain limited. In this work, we develop a comprehensive statistical theory for ERM trees under random design in both high-dimensional regression and classification. We first establish sharp oracle inequalities that bound the excess risk of the ERM estimator relative to the best possible approximation achievable by any tree with at most $L$ leaves, thereby characterizing the interpretability-accuracy trade-off. We derive these results using a novel uniform concentration framework based on empirically localized Rademacher complexity. Furthermore, we derive minimax optimal rates over a novel function class: the piecewise sparse heterogeneous anisotropic Besov (PSHAB) space. This space explicitly captures three key structural features encountered in practice: sparsity, anisotropic smoothness, and spatial heterogeneity. While our main results are established under sub-Gaussianity, we also provide robust guarantees that hold under heavy-tailed noise settings. Together, these findings provide a principled foundation for the optimality of ERM trees and introduce empirical process tools broadly applicable to other highly adaptive, data-driven procedures.

Ruinan Jin, Gexin Huang, Xinwei Shen et al.

Medical imaging diagnosis presents inherent challenges due to diseases that mimic normal anatomy and exhibit significant inter-patient variability. Clinicians routinely employ comparative reasoning-using reference images from healthy controls or previous patient examinations-to discern subtle yet diagnostically critical abnormalities. However, existing medical vision-language models (VLMs) focus primarily on single-image or single-series analyses and lack explicit mechanisms for comparative reasoning. Conversely, general-purpose VLMs demonstrate strong multi-image comparative reasoning capabilities but lack essential medical-domain knowledge to identify nuanced clinical differences. This work aims to bridge this gap by exploring clinically-inspired comparative analysis within VLMs, leveraging reference images to enhance diagnostic accuracy. Through extensive empirical analysis, we show that providing general-purpose VLMs with query and normative matched reference images, accompanied by clinically-informed comparative prompts, significantly improves diagnostic outcomes compared to single-image baselines, especially after supervised finetuning (SFT). Our contributions highlight the clinical relevance of comparative analysis introduce novel strategies for leveraging reference images in VLMs, empirically demonstrate enhanced performance across multiple medical visual question answering (VQA) tasks, and provide theoretical insights into the efficacy of comparative image analysis in medical diagnosis.

Xin Chen, Jason M. Klusowski, Yan Shuo Tan et al.

Combining randomized estimators in an ensemble, such as via random forests, has become a fundamental technique in modern data science, but can be computationally expensive. Furthermore, the mechanism by which this improves predictive performance is poorly understood. We address these issues in the context of sparse linear regression by proposing and analyzing an ensemble of greedy forward selection estimators that are randomized by feature subsampling -- at each iteration, the best feature is selected from within a random subset. We design a novel implementation based on dynamic programming that greatly improves its computational efficiency. Furthermore, we show via careful numerical experiments that our method can outperform popular methods such as lasso and elastic net across a wide range of settings. Next, contrary to prevailing belief that randomized ensembling is analogous to shrinkage, we show via numerical experiments that it can simultaneously reduce training error and degrees of freedom, thereby shifting the entire bias-variance trade-off curve of the base estimator. We prove this fact rigorously in the setting of orthogonal features, in which case, the ensemble estimator rescales the ordinary least squares coefficients with a two-parameter family of logistic weights, thereby enlarging the model search space. These results enhance our understanding of random forests and suggest that implicit regularization in general may have more complicated effects than explicit regularization.

Yan Shuo Tan, Omer Ronen, Theo Saarinen et al.

Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression model that is commonly used in causal inference and beyond. Its strong predictive performance is supported by well-developed estimation theory, comprising guarantees that its posterior distribution concentrates around the true regression function at optimal rates under various data generative settings and for appropriate prior choices. However, the computational properties of the widely-used BART sampler proposed by Chipman et al. (2010) are yet to be well-understood. In this paper, we perform an asymptotic analysis of a slightly modified version of the default BART sampler when fitted to data-generating processes with discrete covariates. We show that the sampler's time to convergence, evaluated in terms of the hitting time of a high posterior density set, increases with the number of training samples, due to the multi-modal nature of the target posterior. On the other hand, we show that this trend can be dampened by simple changes, such as increasing the number of trees in the ensemble or raising the temperature of the sampler. These results provide a nuanced picture on the computational efficiency of the BART sampler in the presence of large amounts of training data while suggesting strategies to improve the sampler. We complement our theoretical analysis with a simulation study focusing on the default BART sampler. We observe that the increasing trend of convergence time against number training samples holds for the default BART sampler and is robust to changes in sampler initialization, number of burn-in iterations, feature selection prior, and discretization strategy. On the other hand, increasing the number of trees or raising the temperature sharply dampens this trend, as indicated by our theory.

Yan Shuo Tan, Chandan Singh, Keyan Nasseri et al.

Modern machine learning has achieved impressive prediction performance, but often sacrifices interpretability, a critical consideration in high-stakes domains such as medicine. In such settings, practitioners often use highly interpretable decision tree models, but these suffer from inductive bias against additive structure. To overcome this bias, we propose Fast Interpretable Greedy-Tree Sums (FIGS), which generalizes the CART algorithm to simultaneously grow a flexible number of trees in summation. By combining logical rules with addition, FIGS is able to adapt to additive structure while remaining highly interpretable. Extensive experiments on real-world datasets show that FIGS achieves state-of-the-art prediction performance. To demonstrate the usefulness of FIGS in high-stakes domains, we adapt FIGS to learn clinical decision instruments (CDIs), which are tools for guiding clinical decision-making. Specifically, we introduce a variant of FIGS known as G-FIGS that accounts for the heterogeneity in medical data. G-FIGS derives CDIs that reflect domain knowledge and enjoy improved specificity (by up to 20% over CART) without sacrificing sensitivity or interpretability. To provide further insight into FIGS, we prove that FIGS learns components of additive models, a property we refer to as disentanglement. Further, we show (under oracle conditions) that unconstrained tree-sum models leverage disentanglement to generalize more efficiently than single decision tree models when fitted to additive regression functions. Finally, to avoid overfitting with an unconstrained number of splits, we develop Bagging-FIGS, an ensemble version of FIGS that borrows the variance reduction techniques of random forests. Bagging-FIGS enjoys competitive performance with random forests and XGBoost on real-world datasets.

Yan Shuo Tan, Abhineet Agarwal, Bin Yu

Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.

Raaz Dwivedi, Yan Shuo Tan, Briton Park et al.

Building on Yu and Kumbier's PCS framework and for randomized experiments, we introduce a novel methodology for Stable Discovery of Interpretable Subgroups via Calibration (StaDISC), with large heterogeneous treatment effects. StaDISC was developed during our re-analysis of the 1999-2000 VIGOR study, an 8076 patient randomized controlled trial (RCT), that compared the risk of adverse events from a then newly approved drug, Rofecoxib (Vioxx), to that from an older drug Naproxen. Vioxx was found to, on average and in comparison to Naproxen, reduce the risk of gastrointestinal (GI) events but increase the risk of thrombotic cardiovascular (CVT) events. Applying StaDISC, we fit 18 popular conditional average treatment effect (CATE) estimators for both outcomes and use calibration to demonstrate their poor global performance. However, they are locally well-calibrated and stable, enabling the identification of patient groups with larger than (estimated) average treatment effects. In fact, StaDISC discovers three clinically interpretable subgroups each for the GI outcome (totaling 29.4% of the study size) and the CVT outcome (totaling 11.0%). Complementary analyses of the found subgroups using the 2001-2004 APPROVe study, a separate independently conducted RCT with 2587 patients, provides further supporting evidence for the promise of StaDISC.

Nick Altieri, Rebecca L. Barter, James Duncan et al.

As the COVID-19 outbreak evolves, accurate forecasting continues to play an extremely important role in informing policy decisions. In this paper, we present our continuous curation of a large data repository containing COVID-19 information from a range of sources. We use this data to develop predictions and corresponding prediction intervals for the short-term trajectory of COVID-19 cumulative death counts at the county-level in the United States up to two weeks ahead. Using data from January 22 to June 20, 2020, we develop and combine multiple forecasts using ensembling techniques, resulting in an ensemble we refer to as Combined Linear and Exponential Predictors (CLEP). Our individual predictors include county-specific exponential and linear predictors, a shared exponential predictor that pools data together across counties, an expanded shared exponential predictor that uses data from neighboring counties, and a demographics-based shared exponential predictor. We use prediction errors from the past five days to assess the uncertainty of our death predictions, resulting in generally-applicable prediction intervals, Maximum (absolute) Error Prediction Intervals (MEPI). MEPI achieves a coverage rate of more than 94% when averaged across counties for predicting cumulative recorded death counts two weeks in the future. Our forecasts are currently being used by the non-profit organization, Response4Life, to determine the medical supply need for individual hospitals and have directly contributed to the distribution of medical supplies across the country. We hope that our forecasts and data repository at https://covidseverity.com can help guide necessary county-specific decision-making and help counties prepare for their continued fight against COVID-19.

Yan Shuo Tan, Roman Vershynin

In recent literature, a general two step procedure has been formulated for solving the problem of phase retrieval. First, a spectral technique is used to obtain a constant-error initial estimate, following which, the estimate is refined to arbitrary precision by first-order optimization of a non-convex loss function. Numerical experiments, however, seem to suggest that simply running the iterative schemes from a random initialization may also lead to convergence, albeit at the cost of slightly higher sample complexity. In this paper, we prove that, in fact, constant step size online stochastic gradient descent (SGD) converges from arbitrary initializations for the non-smooth, non-convex amplitude squared loss objective. In this setting, online SGD is also equivalent to the randomized Kaczmarz algorithm from numerical analysis. Our analysis can easily be generalized to other single index models. It also makes use of new ideas from stochastic process theory, including the notion of a summary state space, which we believe will be of use for the broader field of non-convex optimization.

Yan Shuo Tan

We consider the problem of high-dimensional misspecified phase retrieval. This is where we have an $s$-sparse signal vector $\mathbf{x}_*$ in $\mathbb{R}^n$, which we wish to recover using sampling vectors $\textbf{a}_1,\ldots,\textbf{a}_m$, and measurements $y_1,\ldots,y_m$, which are related by the equation $f(\left<\textbf{a}_i,\textbf{x}_*\right>) = y_i$. Here, $f$ is an unknown link function satisfying a positive correlation with the quadratic function. This problem was analyzed in a recent paper by Neykov, Wang and Liu, who provided recovery guarantees for a two-stage algorithm with sample complexity $m = O(s^2\log n)$. In this paper, we show that the first stage of their algorithm suffices for signal recovery with the same sample complexity, and extend the analysis to non-Gaussian measurements. Furthermore, we show how the algorithm can be generalized to recover a signal vector $\textbf{x}_*$ efficiently given geometric prior information other than sparsity.

John Lipor, David Hong, Yan Shuo Tan et al.

Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees, or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the K-subspaces (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble K-subspaces (EKSS), forms a co-association matrix whose (i,j)th entry is the number of times points i and j are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles, and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace.

Yan Shuo Tan, Roman Vershynin

We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a computational edge over state-of-the-art Wirtinger flow methods. In this paper, we provide the first theoretical guarantee for the convergence of the randomized Kaczmarz method for phase retrieval. We show that it is sufficient to have as many Gaussian measurements as the dimension, up to a constant factor. Along the way, we introduce a sufficient condition on measurement sets for which the randomized Kaczmarz method is guaranteed to work. We show that Gaussian sampling vectors satisfy this property with high probability; this is proved using a chaining argument coupled with bounds on VC dimension and metric entropy.

Yan Shuo Tan, Roman Vershynin

The problem of Non-Gaussian Component Analysis (NGCA) is about finding a maximal low-dimensional subspace $E$ in $\mathbb{R}^n$ so that data points projected onto $E$ follow a non-gaussian distribution. Although this is an appropriate model for some real world data analysis problems, there has been little progress on this problem over the last decade. In this paper, we attempt to address this state of affairs in two ways. First, we give a new characterization of standard gaussian distributions in high-dimensions, which lead to effective tests for non-gaussianness. Second, we propose a simple algorithm, \emph{Reweighted PCA}, as a method for solving the NGCA problem. We prove that for a general unknown non-gaussian distribution, this algorithm recovers at least one direction in $E$, with sample and time complexity depending polynomially on the dimension of the ambient space. We conjecture that the algorithm actually recovers the entire $E$.