$k$k -means clustering is a fundamental problem in many scientific and engineering domains. The optimization problem associated with $k$k-means clustering is nonconvex, for which standard algorithms are only guaranteed to find a local optimum. Leveraging the hidden structure of local solutions, we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution or the ground truth clustering. This framework consists of iteratively alternating between two steps: (i) detect mis-specified clusters in a local solution, and (ii) improve the local solution by non-local operations. We discuss specific implementation of these steps, and elucidate how the proposed framework unifies many existing variants of $k$k-means algorithms through a geometric perspective. We also present two natural variants of the proposed framework, where the initial number of clusters may be over- or under-specified. We provide theoretical justifications and extensive experiments to demonstrate the efficacy of the proposed approach.

This paper investigates a general robust one-shot aggregation framework for distributed and federated Independent Component Analysis (ICA) problem. We propose a geometric median-based aggregation algorithm that leverages $k$-means clustering to resolve the permutation ambiguity in local client estimations. Our method first performs k-means to partition client-provided estimators into clusters and then aggregates estimators within each cluster using the geometric median. This approach provably remains effective even in highly heterogeneous scenarios where at most half of the clients can observe only a minimal number of samples. The key theoretical contribution lies in the combined analysis of the geometric median's error bound-aided by sample quantiles-and the maximum misclustering rates of the aforementioned solution of $k$-means. The effectiveness of the proposed approach is further supported by simulation studies conducted under various heterogeneous settings.

Surgical phase recognition (SPR) is an integral component of surgical data science, enabling high-level surgical analysis. End-to-end trained neural networks that predict the surgical phase directly from videos have shown excellent performance on benchmarks. However, these models struggle with robustness due to non-causal associations in the training set. Our goal is to improve model robustness to variations in the surgical videos by leveraging the digital twin (DT) paradigm – an intermediary layer to separate high-level analysis (SPR) from low-level processing. As a proof of concept, we present a DT representation-based framework for SPR from videos. The framework employs vision foundation models with reliable low-level scene understanding to craft DT representation. We embed the DT representation in place of raw video inputs in the state-of-the-art SPR model. The framework is trained on the Cholec80 dataset and evaluated on out-of-distribution (OOD) and corrupted test samples. Contrary to the vulnerability of the baseline model, our framework demonstrates strong robustness on both OOD and corrupted samples, with a video-level accuracy of 80.3 on a highly corrupted Cholec80 test set, 67.9 on the challenging CRCD dataset, and 99.8 on an internal robotic surgery dataset, outperforming the baseline by 3.9, 16.8, and 90.9 respectively. We also find that using DT representation as an augmentation to the raw input can significantly improve model robustness. Our findings lend support to the thesis that DT representations are effective in enhancing model robustness. Future work will seek to improve the feature informativeness and incorporate interpretability for a more comprehensive framework.

The integration of network information and node attribute information has recently gained significant attention in the community detection literature. In this work, we consider community detection in the Contextual Labeled Stochastic Block Model (CLSBM), where the network follows an LSBM and node attributes follow a Gaussian Mixture Model (GMM). Our primary focus is the misclassification rate, which measures the expected number of nodes misclassified by community detection algorithms. We first establish a lower bound on the optimal misclassification rate that holds for any algorithm. When we specialize our setting to the LSBM (which preserves only network information) or the GMM (which preserves only node attribute information), our lower bound recovers prior results. Moreover, we present an efficient spectral-based algorithm tailored for the CLSBM and derive an upper bound on its misclassification rate. Although the algorithm does not attain the lower bound, it serves as a reliable starting point for designing more accurate community detection algorithms (as many algorithms use spectral method as an initial step, followed by refinement procedures to enhance accuracy).

Large language models (LLMs) have demonstrated remarkable capabilities across various tasks. However, these models could offer biased, hallucinated, or non-factual responses camouflaged by their fluency and realistic appearance. Uncertainty estimation is the key method to address this challenge. While research efforts in uncertainty estimation are ramping up, there is a lack of comprehensive and dedicated surveys on LLM uncertainty estimation. This survey presents four major avenues of LLM uncertainty estimation. Furthermore, we perform extensive experimental evaluations across multiple methods and datasets. At last, we provide critical and promising future directions for LLM uncertainty estimation.

Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. The most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided to date mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices. In this paper, we propose a deflation varimax procedure that solves each row of an orthogonal matrix sequentially. In addition to its net computational gain and flexibility, we are able to fully establish theoretical guarantees for the proposed procedure in a broader context. Adopting this new deflation varimax as the second step after PCA, we further analyze this two step procedure under a general class of factor models. Our results show that it estimates the factor loading matrix in the minimax optimal rate when the signal-to-noise-ratio (SNR) is moderate or large. In the low SNR regime, we offer possible improvement over using PCA and the deflation varimax when the additive noise under the factor model is structured. The modified procedure is shown to be minimax optimal in all SNR regimes. Our theory is valid for finite sample and allows the number of the latent factors to grow with the sample size as well as the ambient dimension to grow with, or even exceed, the sample size. Extensive simulation and real data analysis further corroborate our theoretical findings.

Federated learning has recently garnered significant attention, especially within the domain of supervised learning. However, despite the abundance of unlabeled data on end-users, unsupervised learning problems such as clustering in the federated setting remain underexplored. In this paper, we investigate the federated clustering problem, with a focus on federated k-means. We outline the challenge posed by its non-convex objective and data heterogeneity in the federated framework. To tackle these challenges, we adopt a new perspective by studying the structures of local solutions in k-means and propose a one-shot algorithm called FeCA (Federated Centroid Aggregation). FeCA adaptively refines local solutions on clients, then aggregates these refined client solutions to recover the global solution of the entire dataset in a single round. We empirically demonstrate the robustness of FeCA under various federated scenarios on both synthetic and real-world data. Additionally, we extend FeCA to representation learning and present DeepFeCA, which combines DeepCluster and FeCA for unsupervised feature learning in the federated setting.

Surgical data science has seen rapid advancement with the excellent performance of end-to-end deep neural networks (DNNs). Despite their successes, DNNs have been proven susceptible to minor "corruptions," introducing a major concern for the translation of cutting-edge technology, especially in high-stakes scenarios. We introduce the SegSTRONG-C challenge dedicated to better understanding model deterioration under unforeseen but plausible non-adversarial "corruption" and the capabilities of contemporary methods that seek to improve it. Built on a dataset generated through counterfactual robotic replay, SegSTRONG-C provides paired clean and "corrupted" samples, enabling reproducible evaluation of model robustness. Participants are challenged to train tool segmentation algorithms on "uncorrupted" data and evaluate them on "corrupted" test domains for the binary robot tool segmentation task. Through comprehensive baseline experiments and participating submissions from widespread community engagement, SegSTRONG-C reveals key themes for model failure and identifies promising directions for improving robustness. The performance of challenge winners, achieving an average 0.9394 DSC and 0.9301 NSD across the unreleased test sets with "corruption" types: bleeding, smoke, and low brightness. This highlights how prior knowledge, customized training strategies, and architectural choice can be leveraged to improve robustness. In conclusion, the SegSTRONG-C challenge has identified practical approaches for enhancing model robustness. However, most approaches rely on conventional techniques that have known limitations. Looking ahead, we advocate for expanding intellectual diversity and creativity in non-adversarial robustness beyond data augmentation, calling for new paradigms that enhance universal robustness to unforeseen "corruptions" to facilitate richer applications in surgical data science.

We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can exist multiple spurious local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: (1) fitting a single mean estimate to multiple Gaussian components or (2) fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation parameter.

Recent advances in Large Language Models (LLMs) have showcased their remarkable reasoning capabilities, making them influential across various fields. However, in robotics, their use has primarily been limited to manipulation planning tasks due to their inherent textual output. This paper addresses this limitation by investigating the potential of adopting the reasoning ability of LLMs for generating numerical predictions in robotics tasks, specifically for robotic grasping. We propose Reasoning Tuning, a novel method that integrates a reasoning phase before prediction during training, leveraging the extensive prior knowledge and advanced reasoning abilities of LLMs. This approach enables LLMs, notably with multi-modal capabilities, to generate accurate numerical outputs like grasp poses that are context-aware and adaptable through conversations. Additionally, we present the Reasoning Tuning VLM Grasp dataset, carefully curated to facilitate the adaptation of LLMs to robotic grasping. Extensive validation on both grasping datasets and real-world experiments underscores the adaptability of multi-modal LLMs for numerical prediction tasks in robotics. This not only expands their applicability but also bridges the gap between text-based planning and direct robot control, thereby maximizing the potential of LLMs in robotics. Our dataset will be released. More details and videos of this work are available on our project page: https://sites.google.com/view/rt-grasp.

$k$-means clustering is a fundamental problem in various disciplines. This problem is nonconvex, and standard algorithms are only guaranteed to find a local optimum. Leveraging the structure of local solutions characterized in [1], we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution (or the ground truth). This framework consists of alternating between the following two steps iteratively: (i) detect mis-specified clusters in a local solution and (ii) improve the current local solution by non-local operations. We discuss implementation of these steps, and elucidate how the proposed framework unifies variants of $k$-means algorithm in literature from a geometric perspective. In addition, we introduce two natural extensions of the proposed framework, where the initial number of clusters is misspecified. We provide theoretical justification for our approach, which is corroborated with extensive experiments.

The $k\text {-means}$ clustering problem concerns finding a partition of the data points into $k$ clusters such that the total within-cluster squared distance is minimized. This optimization objective is non-convex, and not everywhere differentiable. In general, there exist spurious local solutions other than the global optimum. Moreover, the simplest and most popular algorithm for $k\text {-means}$ , namely Lloyd's algorithm, generally converges to such spurious local solutions both in theory and in practice. In this paper, we investigate the structures of these spurious local solutions under a probabilistic generative model with $k$ ground truth clusters. As soon as $k=3$ , spurious local minima provably exist, even for well-separated clusters. One such local minimum puts two centers at one true cluster, and the third center in the middle of the other two true clusters. We prove that this is essentially the only type of spurious local minima under a separation condition. In particular, any local minimum solution only involves a configuration that puts multiple centers at a true cluster, and one center in the middle of multiple true clusters. Our results pertain to the $k\text {-means}$ formulation for mixtures of Gaussians or bounded distributions, and hold in the over- and under-parametrization regimes where the number of centers in $k\text {-means}$ may not equal to the number of true clusters. Our theoretical results corroborate existing empirical observations and provide justification for popular heuristics for $k\text {-means}$ clustering.

Blind deconvolution is the problem of recovering a convolutional kernel and an activation signal from their convolution y = a0 * x0. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed.

The problem of finding a unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $\boldsymbol Y\in \mathbb R ^{p\times n}$ that admits a sparse representation. Specifically, we consider $\boldsymbol Y= \boldsymbol A \boldsymbol X $ where the matrix $\boldsymbol A\in \mathbb R^{p\times r}$ has full column rank, with $r < \min \{n,p\}$ , and the matrix $\boldsymbol X\in \mathbb R^{r\times n}$ is element-wise sparse. We prove that this low rank, sparse decomposition of $\boldsymbol Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for its nonconvex optimization landscape shows that any strict local solution is close to the ground truth, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. Our theoretical findings are corroborated by numerical experiments.1

Learning overcomplete representations finds many applications in machine learning and data analytics. In the past decade, despite the empirical success of heuristic methods, theoretical understandings and explanations of these algorithms are still far from satisfactory. In this work, we provide new theoretical insights for several important representation learning problems: learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries. We formulate these problems as $\ell^4$-norm optimization problems over the sphere and study the geometric properties of their nonconvex optimization landscapes. For both problems, we show the nonconvex objective has benign (global) geometric structures, which enable the development of efficient optimization methods finding the target solutions. Finally, our theoretical results are justified by numerical simulations.

Blind deconvolution is a ubiquitous problem of recovering two unknown signals from their convolution. Unfortunately, this is an ill-posed problem in general. This paper focuses on the short and sparse blind deconvolution problem, where the one unknown signal is short and the other one is sparsely and randomly supported. This variant captures the structure of the unknown signals in several important applications. We assume the short signal to have unit ℓ^2 norm and cast the blind deconvolution problem as a nonconvex optimization problem over the sphere. We demonstrate that (i) in a certain region of the sphere, every local optimum is close to some shift truncation of the ground truth, and (ii) for a generic short signal of length k, when the sparsity of activation signal θ≲ k^-2/3 and number of measurements m≳ poly(k), a simple initialization method together with a descent algorithm which escapes strict saddle points recovers a near shift truncation of the ground truth kernel.

Modern high-resolution microscopes, such as the scanning tunneling microscope, are commonly used to study specimens that have dense and aperiodic spatial structure. Extracting meaningful information from images obtained from such microscopes remains a formidable challenge. Fourier analysis is commonly used to analyze the underlying structure of fundamental motifs present in an image. However, the Fourier transform fundamentally suffers from severe phase noise when applied to aperiodic images. Here, we report the development of a new algorithm based on nonconvex optimization, applicable to any microscopy modality, that directly uncovers the fundamental motifs present in a real-space image. Apart from being quantitatively superior to traditional Fourier analysis, we show that this novel algorithm also uncovers phase sensitive information about the underlying motif structure. We demonstrate its usefulness by studying scanning tunneling microscopy images of a Co-doped iron arsenide superconductor and prove that the application of the algorithm allows for the complete recovery of quasiparticle interference in this material. Our phase sensitive quasiparticle interference imaging results indicate that the pairing symmetry in optimally doped NaFeAs is consistent with a sign-changing s+- order parameter.

This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and wish to recover the missing entries under this assumption. We show that we can identify the complete matrix of minimum intrinsic dimension by minimizing the rank of the matrix in a high dimensional feature space. We develop a new formulation of the resulting problem using the kernel trick together with a new relaxation of the rank objective, and propose an efficient optimization method. We also show how to use our methods to complete data drawn from multiple nonlinear manifolds. Comparative studies on synthetic data, subspace clustering with missing data, motion capture data recovery, and transductive learning verify the superiority of our methods over the state-of-the-art.

Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing global and fast convergence for these algorithms. In this paper, we show that a critical component in provable low-rank recovery with non-quadratic loss is a regularity projection oracle. This oracle restricts iterates to low-rank matrices within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of approximate RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic low-rank estimation problems including one bit matrix sensing/completion, individualized rank aggregation, and more broadly generalized linear models with rank constraints.

As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from signal and data acquisition to modeling and prediction. The optimization problems encountered in practice are often nonconvex. While challenges vary from problem to problem, one common source of nonconvexity is nonlinearity in the data or measurement model. Nonlinear models often exhibit symmetries, creating complicated, nonconvex objective landscapes, with multiple equivalent solutions. Nevertheless, simple methods (e.g., gradient descent) often perform surprisingly well in practice. The goal of this survey is to highlight a class of tractable nonconvex problems, which can be understood through the lens of symmetries. These problems exhibit a characteristic geometric structure: local minimizers are symmetric copies of a single "ground truth" solution, while other critical points occur at balanced superpositions of symmetric copies of the ground truth, and exhibit negative curvature in directions that break the symmetry. This structure enables efficient methods to obtain global minimizers. We discuss examples of this phenomenon arising from a wide range of problems in imaging, signal processing, and data analysis. We highlight the key role of symmetry in shaping the objective landscape and discuss the different roles of rotational and discrete symmetries. This area is rich with observed phenomena and open problems; we close by highlighting directions for future research.