Despite the fact that subspace clustering has become a powerful Given a trimmed sequence, in which a single action or activity technique for problems such as face clustering or digit is assumed to be present, the final goal of HAR is to correctly recognition, its applicability to the problems like skeletonbased classifying it. Although significant progresses have been made HAR was only explored by a limited number of works in the last years, accurate action recognition in videos is still a [7], [8], [9]. This is due to many operative limitations including challenging task because of the complexity of the visual data how to handle the temporal dimensions, the inherent noise e.g., due to varying camera viewpoints, occlusions and abrupt present in the skeletal data and the related computational changes in lighting conditions.

What does it mean for a clustering to be fair? One popular approach seeks to ensure that each cluster contains groups in (roughly) the same proportion in which they exist in the population. The normative principle at play is balance: any cluster might act as a representative of the data, and thus should reflect its diversity. But clustering also captures a different form of representativeness. A core principle in most clustering problems is that a cluster center should be representative of the cluster it represents, by being "close" to the points associated with it. This is so that we can effectively replace the points by their cluster centers without significant loss in fidelity, and indeed is a common "use case" for clustering. For such a clustering to be fair, the centers should "represent" different groups equally well. We call such a clustering a group-representative clustering. In this paper, we study the structure and computation of group-representative clusterings. We show that this notion naturally parallels the development of fairness notions in classification, with direct analogs of ideas like demographic parity and equal opportunity. We demonstrate how these notions are distinct from and cannot be captured by balance-based notions of fairness. We present approximation algorithms for group representative $k$-median clustering and couple this with an empirical evaluation on various real-world data sets.

Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al.\ \cite{hocking}, ADMM and ADA by Chi and Lange\ \cite{Chi}, stochastic incremental algorithm by Panahi et al.\ \cite{Panahi} and semismooth Newton-CG augmented Lagrangian method by Yuan et al.\ \cite{dsun1}. All algorithms yield approximate solutions, even though an exact solution is demanded to determine the correct cluster assignment. The purpose of this paper is to close the gap between the output from existing algorithms and the exact solution to the optimization problem. We present a clustering test which identifies and certifies the correct cluster assignment from an approximate solution yielded by any primal-dual algorithm. The test may not succeed if the approximation is inaccurate. However, we show the correct cluster assignment is guaranteed to be found by a symmetric primal-dual path following algorithm after sufficiently many iterations, provided that the model parameter $\lambda$ avoids a finite number of bad values. Numerical experiments are implemented to support our results.

Graph neural networks have attracted wide attentions to enable representation learning of graph data in recent works. In complement to graph convolution operators, graph pooling is crucial for extracting hierarchical representation of graph data. However, most recent graph pooling methods still fail to efficiently exploit the geometry of graph data. In this paper, we propose a novel graph pooling strategy that leverages node proximity to improve the hierarchical representation learning of graph data with their multi-hop topology. Node proximity is obtained by harmonizing the kernel representation of topology information and node features. Implicit structure-aware kernel representation of topology information allows efficient graph pooling without explicit eigendecomposition of the graph Laplacian. Similarities of node signals are adaptively evaluated with the combination of the affine transformation and kernel trick using the Gaussian RBF function. Experimental results demonstrate that the proposed graph pooling strategy is able to achieve state-of-the-art performance on a collection of public graph classification benchmark datasets.

Conditional correlation networks, within Gaussian Graphical Models (GGM), are widely used to describe the direct interactions between the components of a random vector. In the case of an unlabelled Heterogeneous population, Expectation Maximisation (EM) algorithms for Mixtures of GGM have been proposed to estimate both each sub-population's graph and the class labels. However, we argue that, with most real data, class affiliation cannot be described with a Mixture of Gaussian, which mostly groups data points according to their geometrical proximity. In particular, there often exists external co-features whose values affect the features' average value, scattering across the feature space data points belonging to the same sub-population. Additionally, if the co-features' effect on the features is Heterogeneous, then the estimation of this effect cannot be separated from the sub-population identification. In this article, we propose a Mixture of Conditional GGM (CGGM) that subtracts the heterogeneous effects of the co-features to regroup the data points into sub-population corresponding clusters. We develop a penalised EM algorithm to estimate graph-sparse model parameters. We demonstrate on synthetic and real data how this method fulfils its goal and succeeds in identifying the sub-populations where the Mixtures of GGM are disrupted by the effect of the co-features.

In clustering problems, a central decision-maker is given a complete metric graph over vertices and must provide a clustering of vertices that minimizes some objective function. In fair clustering problems, vertices are endowed with a color (e.g., membership in a group), and the features of a valid clustering might also include the representation of colors in that clustering. Prior work in fair clustering assumes complete knowledge of group membership. In this paper, we generalize prior work by assuming imperfect knowledge of group membership through probabilistic assignments. We present clustering algorithms in this more general setting with approximation ratio guarantees. We also address the problem of "metric membership", where different groups have a notion of order and distance. Experiments are conducted using our proposed algorithms as well as baselines to validate our approach and also surface nuanced concerns when group membership is not known deterministically.

Robust Subspace Recovery (RoSuRe) algorithm was recently introduced as a principled and numerically efficient algorithm that unfolds underlying Unions of Subspaces (UoS) structure, present in the data. The union of Subspaces (UoS) is capable of identifying more complex trends in data sets than simple linear models. We build on and extend RoSuRe to prospect the structure of different data modalities individually. We propose a novel multi-modal data fusion approach based on group sparsity which we refer to as Robust Group Subspace Recovery (RoGSuRe). Relying on a bi-sparsity pursuit paradigm and non-smooth optimization techniques, the introduced framework learns a new joint representation of the time series from different data modalities, respecting an underlying UoS model. We subsequently integrate the obtained structures to form a unified subspace structure. The proposed approach exploits the structural dependencies between the different modalities data to cluster the associated target objects. The resulting fusion of the unlabeled sensors' data from experiments on audio and magnetic data has shown that our method is competitive with other state of the art subspace clustering methods. The resulting UoS structure is employed to classify newly observed data points, highlighting the abstraction capacity of the proposed method.

As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find a fair hierarchical clustering, with only a negligible loss in the objective.

Robust representation learning of temporal dynamic interactions is an important problem in robotic learning in general and automated unsupervised learning in particular. Temporal dynamic interactions can be described by (multiple) geometric trajectories in a suitable space over which unsupervised learning techniques may be applied to extract useful features from raw and high-dimensional data measurements. Taking a geometric approach to robust representation learning for temporal dynamic interactions, it is necessary to develop suitable metrics and a systematic methodology for comparison and for assessing the stability of an unsupervised learning method with respect to its tuning parameters. Such metrics must account for the (geometric) constraints in the physical world as well as the uncertainty associated with the learned patterns. In this paper we introduce a model-free metric based on the Procrustes distance for robust representation learning of interactions, and an optimal transport based distance metric for comparing between distributions of interaction primitives. These distance metrics can serve as an objective for assessing the stability of an interaction learning algorithm. They are also used for comparing the outcomes produced by different algorithms. Moreover, they may also be adopted as an objective function to obtain clusters and representative interaction primitives. These concepts and techniques will be introduced, along with mathematical properties, while their usefulness will be demonstrated in unsupervised learning of vehicle-to-vechicle interactions extracted from the Safety Pilot database, the world's largest database for connected vehicles.

We show that the popular $k$-means clustering algorithm (Lloyd's heuristic), used for a variety of scientific data, can result in outcomes that are unfavorable to subgroups of data (e.g., demographic groups). Such biased clusterings can have deleterious implications for human-centric applications such as resource allocation. We present a fair $k$-means objective and algorithm to choose cluster centers that provide equitable costs for different groups. The algorithm, Fair-Lloyd, is a modification of Lloyd's heuristic for $k$-means, inheriting its simplicity, efficiency, and stability. In comparison with standard Lloyd's, we find that on benchmark data sets, Fair-Lloyd exhibits unbiased performance by ensuring that all groups have balanced costs in the output $k$-clustering, while incurring a negligible increase in running time, thus making it a viable fair option wherever $k$-means is currently used.