Clustering and dimensionality reduction have been crucial topics in machine learning and computer vision. Clustering high-dimensional data has been challenging for a long time due to the curse of dimensionality. For that reason, a more promising direction is the joint learning of dimension reduction and clustering. In this work, we propose a Manifold Learning Framework that learns dimensionality reduction and clustering simultaneously. The proposed framework is able to jointly learn the parameters of a dimension reduction technique (e.g. linear projection or a neural network) and cluster the data based on the resulting features (e.g. under a Gaussian Mixture Model framework). The framework searches for the dimension reduction parameters and the optimal clusters by traversing a manifold,using Gradient Manifold Optimization. The obtained The proposed framework is exemplified with a Gaussian Mixture Model as one simple but efficient example, in a process that is somehow similar to unsupervised Linear Discriminant Analysis (LDA). We apply the proposed method to the unsupervised training of simulated data as well as a benchmark image dataset (i.e. MNIST). The experimental results indicate that our algorithm has better performance than popular clustering algorithms from the literature.

Demographic parity (DP) is a widely studied fairness criterion in regression, enforcing independence between the predictions and sensitive attributes. However, constraining the entire distribution can degrade predictive accuracy and may be unnecessary for many applications, where fairness concerns are localized to specific regions of the distribution. To overcome this issue, we propose a new framework for regression under DP that focuses on the tails of target distribution across sensitive groups. Our methodology builds on optimal transport theory. By enforcing fairness constraints only over targeted regions of the distribution, our approach enables more nuanced and context-sensitive interventions. Leveraging recent advances, we develop an interpretable and flexible algorithm that leverages the geometric structure of optimal transport. We provide theoretical guarantees, including risk bounds and fairness properties, and validate the method through experiments in regression settings.

Constrained optimization in high-dimensional black-box settings is difficult due to expensive evaluations, the lack of gradient information, and complex feasibility regions. In this work, we propose a Bayesian optimization method that combines a penalty formulation, a surrogate model, and a trust region strategy. The constrained problem is converted to an unconstrained form by penalizing constraint violations, which provides a unified modeling framework. A trust region restricts the search to a local region around the current best solution, which improves stability and efficiency in high dimensions. Within this region, we use the Expected Improvement acquisition function to select evaluation points by balancing improvement and uncertainty. The proposed Trust Region method integrates penalty-based constraint handling with local surrogate modeling. This combination enables efficient exploration of feasible regions while maintaining sample efficiency. We compare the proposed method with state-of-the-art methods on synthetic and real-world high-dimensional constrained optimization problems. The results show that the method identifies high-quality feasible solutions with fewer evaluations and maintains stable performance across different settings.

Recently, several works have shown that natural modifications of the classical conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex optimization, provably converge with a linear rate when the feasible set is a polytope, and the objective is smooth and strongly-convex. However, all of these results suffer from two significant shortcomings: i) large memory requirement due to the need to store an explicit convex decomposition of the current iterate, and as a consequence, large running-time overhead per iteration ii) the worst case convergence rate depends unfavorably on the dimension In this work we present a new conditional gradient variant and a corresponding analysis that improves on both of the above shortcomings. In particular, both memory and computation overheads are only linear in the dimension, and in addition, in case the optimal solution is sparse, the new convergence rate replaces a factor which is at least linear in the dimension in previous works, with a linear dependence on the number of non-zeros in the optimal solution At the heart of our method, and corresponding analysis, is a novel way to compute decomposition-invariant away-steps. While our theoretical guarantees do not apply to any polytope, they apply to several important structured polytopes that capture central concepts such as paths in graphs, perfect matchings in bipartite graphs, marginal distributions that arise in structured prediction tasks, and more. Our theoretical findings are complemented by empirical evidence that shows that our method delivers state-of-the-art performance.

We define and study the problem of predicting the solution to a linear program (LP) given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, that has been studied under the name "Learning from Revealed Preferences. We give mistake bound learning algorithms in two settings: in the first, the objective of the LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown. In the second setting, the objective of the LP is unknown, and changing in a controlled way. The constraints of the LP may also change every day, but are known. An example is given by a set of constraints and partial information about the objective, and the task of the learner is again to predict the optimal solution of the partially known LP.

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem.

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Author is also affiliated with RaySearch Laboratories. Here is an incomplete list. Our proof consist of two parts. There are two cases to consider. There are two cases to consider.