Karan Srivastava, Daniel L. Pimentel-Alarcón
This paper derives a perturbation bound on the optimal subspace estimator obtained from a subset of its canonical projections contaminated by noise. This fundamental result has important implications in matrix completion, subspace clustering, and related problems.
Usman Mahmood, Daniel Pimentel-Alarcón
This paper introduces {\em fusion subspace clustering}, a novel method to learn low-dimensional structures that approximate large scale yet highly incomplete data. The main idea is to assign each datum to a subspace of its own, and minimize the distance between the subspaces of all data, so that subspaces of the same cluster get {\em fused} together. Our method allows low, high, and even full-rank data; it directly accounts for noise, and its sample complexity approaches the information-theoretic limit. In addition, our approach provides a natural model selection {\em clusterpath}, and a direct completion method. We give convergence guarantees, analyze computational complexity, and show through extensive experiments on real and synthetic data that our approach performs comparably to the state-of-the-art with complete data, and dramatically better if data is missing.
Yuren Sun, Tatiana Midori Maeda, Claudia Solis-Lemus et al.
To protect tropical forest biodiversity, we need to be able to detect it reliably, cheaply, and at scale. Automated species detection from passively recorded soundscapes via machine-learning approaches is a promising technique towards this goal, but it is constrained by the necessity of large training data sets. Using soundscapes from a tropical forest in Borneo and a Convolutional Neural Network model (CNN) created with transfer learning, we investigate i) the minimum viable training data set size for accurate prediction of call types ('sonotypes'), and ii) the extent to which data augmentation can overcome the issue of small training data sets. We found that even relatively high sample sizes (> 80 per call type) lead to mediocre accuracy, which however improves significantly with data augmentation, including at extremely small sample sizes, regardless of taxonomic group or call characteristics. Our results suggest that transfer learning and data augmentation can make the use of CNNs to classify species' vocalizations feasible even for small soundscape-based projects with many rare species. Retraining our open-source model requires only basic programming skills which makes it possible for individual conservation initiatives to match their local context, in order to enable more evidence-informed management of biodiversity.
Huanran Li, Daniel Pimentel-Alarcón
Subspace clustering aims to group data points that lie in a union of low-dimensional subspaces and finds wide application in computer vision, hyperspectral imaging, and recommendation systems. However, most existing methods assume fully observed data, limiting their effectiveness in real-world scenarios with missing entries. In this paper, we propose a contrastive self-supervised framework, Contrastive Subspace Clustering (CSC), designed for clustering incomplete data. CSC generates masked views of partially observed inputs and trains a deep neural network using a SimCLR-style contrastive loss to learn invariant embeddings. These embeddings are then clustered using sparse subspace clustering. Experiments on six benchmark datasets show that CSC consistently outperforms both classical and deep learning baselines, demonstrating strong robustness to missing data and scalability to large datasets.
Huanran Li, Jeremy Johnson, Daniel Pimentel-Alarcón
This paper approaches high-rank matrix completion (HRMC) by filling missing entries in a data matrix where columns lie near a union of subspaces, clustering these columns, and identifying the underlying subspaces. Current methods often lack theoretical support, produce uninterpretable results, and require more samples than theoretically necessary. We propose clustering incomplete vectors by grouping proxy subspaces and minimizing two criteria over the Grassmannian: (a) the chordal distance between each point and its corresponding subspace and (b) the geodesic distances between subspaces of all data points. Experiments on synthetic and real datasets demonstrate that our method performs comparably to leading methods in high sampling rates and significantly better in low sampling rates, thus narrowing the gap to the theoretical sampling limit of HRMC.
Huanran Li, Daniel Pimentel-Alarcón
Contrastive Learning (CL) has emerged as a powerful method for training feature extraction models using unlabeled data. Recent studies suggest that incorporating a linear projection head post-backbone significantly enhances model performance. In this work, we investigate the use of a transformer model as a projection head within the CL framework, aiming to exploit the transformer's capacity for capturing long-range dependencies across embeddings to further improve performance. Our key contributions are fourfold: First, we introduce a novel application of transformers in the projection head role for contrastive learning, marking the first endeavor of its kind. Second, our experiments reveal a compelling "Deep Fusion" phenomenon where the attention mechanism progressively captures the correct relational dependencies among samples from the same class in deeper layers. Third, we provide a theoretical framework that explains and supports this "Deep Fusion" behavior. Finally, we demonstrate through experimental results that our model achieves superior performance compared to the existing approach of using a feed-forward layer.
Heguang Lin, Mengze Li, Daniel Pimentel-Alarcón et al.
Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. This paper studies the geometry of the minimum-volume confidence sets for the multinomial parameter. When used in place of more standard confidence sets and intervals based on bounds and asymptotic approximation, learning algorithms can exhibit improved sample complexity. Prior work showed the minimum-volume confidence sets are the level-sets of a discontinuous function defined by an exact p-value. While the confidence sets are optimal in that they have minimum average volume, computation of membership of a single point in the set is challenging for problems of modest size. Since the confidence sets are level-sets of discontinuous functions, little is apparent about their geometry. This paper studies the geometry of the minimum volume confidence sets by enumerating and covering the continuous regions of the exact p-value function. This addresses a fundamental question in A/B testing: given two multinomial outcomes, how can one determine if their corresponding minimum volume confidence sets are disjoint? We answer this question in a restricted setting.
Md Mahfuzur Rahman, Daniel Pimentel-Alarcon
Due to diverse nature of data acquisition and modern applications, many contemporary problems involve high dimensional datum $\x \in \R^\d$ whose entries often lie in a union of subspaces and the goal is to find out which entries of $\x$ match with a particular subspace $\sU$, classically called \emph {matched subspace detection}. Consequently, entries that match with one subspace are considered as inliers w.r.t the subspace while all other entries are considered as outliers. Proportion of outliers relative to each subspace varies based on the degree of coordinates from subspaces. This problem is a combinatorial NP-hard in nature and has been immensely studied in recent years. Existing approaches can solve the problem when outliers are sparse. However, if outliers are abundant or in other words if $\x$ contains coordinates from a fair amount of subspaces, this problem can't be solved with acceptable accuracy or within a reasonable amount of time. This paper proposes a two-stage approach called \emph{Greedy Linear Integer Mixed Programmed Selector} (GLIMPS) for this abundant-outliers setting, which combines a greedy algorithm and mixed integer formulation and can tolerate over 80\% outliers, outperforming the state-of-the-art.
Daniel L. Pimentel-Alarcón, Usman Mahmood
Modern inference and learning often hinge on identifying low-dimensional structures that approximate large scale data. Subspace clustering achieves this through a union of linear subspaces. However, in contemporary applications data is increasingly often incomplete, rendering standard (full-data) methods inapplicable. On the other hand, existing incomplete-data methods present major drawbacks, like lifting an already high-dimensional problem, or requiring a super polynomial number of samples. Motivated by this, we introduce a new subspace clustering algorithm inspired by fusion penalties. The main idea is to permanently assign each datum to a subspace of its own, and minimize the distance between the subspaces of all data, so that subspaces of the same cluster get fused together. Our approach is entirely new to both, full and missing data, and unlike other methods, it directly allows noise, it requires no liftings, it allows low, high, and even full-rank data, it approaches optimal (information-theoretic) sampling rates, and it does not rely on other methods such as low-rank matrix completion to handle missing data. Furthermore, our extensive experiments on both real and synthetic data show that our approach performs comparably to the state-of-the-art with complete data, and dramatically better if data is missing.
Daniel L. Pimentel-Alarcón
Completing a data matrix X has become an ubiquitous problem in modern data science, with applications in recommender systems, computer vision, and networks inference, to name a few. One typical assumption is that X is low-rank. A more general model assumes that each column of X corresponds to one of several low-rank matrices. This paper generalizes these models to what we call mixture matrix completion (MMC): the case where each entry of X corresponds to one of several low-rank matrices. MMC is a more accurate model for recommender systems, and brings more flexibility to other completion and clustering problems. We make four fundamental contributions about this new model. First, we show that MMC is theoretically possible (well-posed). Second, we give its precise information-theoretic identifiability conditions. Third, we derive the sample complexity of MMC. Finally, we give a practical algorithm for MMC with performance comparable to the state-of-the-art for simpler related problems, both on synthetic and real data.
Greg Ongie, Daniel Pimentel-Alarcón, Laura Balzano et al.
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We provide a formal mathematical justification for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model.
Daniel L. Pimentel-Alarcón, Nigel Boston, Robert D. Nowak
Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable if there are at most finitely many rank-$r$ matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a deterministic sampling condition for finite completability. We use this to also derive deterministic sampling conditions for unique completability that can be efficiently verified. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if $O(\max\{r,\log d\})$ entries per column are observed. These findings have several implications on LRMC regarding lower bounds, sample and computational complexity, the role of coherence, adaptive settings and the validation of any completion algorithm. We complement our theoretical results with experiments that support our findings and motivate future analysis of uncharted sampling regimes.
Daniel L. Pimentel-Alarcón, Robert D. Nowak, Nigel Boston
Consider a generic $r$-dimensional subspace of $\mathbb{R}^d$, $r<d$, and suppose that we are only given projections of this subspace onto small subsets of the canonical coordinates. The paper establishes necessary and sufficient deterministic conditions on the subsets for subspace identifiability.
Daniel L. Pimentel-Alarcón
Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data from a union of subspaces, it is because such data really lies in a subspace. Furthermore, Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data, such subspace is unique. Do this by characterizing when and only when a set of incomplete vectors behaves as a single but complete one.