Universal Domain Adaptation (UniDA) aims to transfer knowledge from a labeled source domain to an unlabeled target domain, even when their classes are not fully shared. Few dedicated UniDA methods exist for Time Series (TS), which remains a challenging case. In general, UniDA approaches align common class samples and detect unknown target samples from emerging classes. Such detection often results from thresholding a discriminability metric. The threshold value is typically either a fine-tuned hyperparameter or a fixed value, which limits the ability of the model to adapt to new data. Furthermore, discriminability metrics exhibit overconfidence for unknown samples, leading to misclassifications. This paper introduces UniJDOT, an optimal-transport-based method that accounts for the unknown target samples in the transport cost. Our method also proposes a joint decision space to improve the discriminability of the detection module. In addition, we use an auto-thresholding algorithm to reduce the dependence on fixed or fine-tuned thresholds. Finally, we rely on a Fourier transform-based layer inspired by the Fourier Neural Operator for better TS representation. Experiments on TS benchmarks demonstrate the discriminability, robustness, and state-of-the-art performance of UniJDOT.

Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computationaland statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.

Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown, in applications such as domain adaptation, to provide performances similar to its non-private (non-smoothed) counterpart. However, the computational and statistical properties of such a metric is not yet been well-established. In this paper, we analyze the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian smoothed sliced divergences. We show that smoothing and slicing preserve the metric property and the weak topology. We also provide results on the sample complexity of such divergences. Since, the privacy level depends on the amount of Gaussian smoothing, we analyze the impact of this parameter on the divergence. We support our theoretical findings with empirical studies of Gaussian smoothed and sliced version of Wassertein distance, Sinkhorn divergence and maximum mean discrepancy (MMD). In the context of privacy-preserving domain adaptation, we confirm that those Gaussian smoothed sliced Wasserstein and MMD divergences perform very well while ensuring data privacy.

This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear regression problem. This reformulation allows us to propose novel algorithms inspired from inverse problems and nonnegative matrix factorization. In particular, we consider majorization-minimization which leads in our setting to efficient multiplicative updates for a variety of penalties. Furthermore, we derive for the first time an efficient algorithm to compute the regularization path of UOT with quadratic penalties. The proposed algorithm provides a continuity of piece-wise linear OT plans converging to the solution of balanced OT (corresponding to infinite penalty weights). We perform several numerical experiments on simulated and real data illustrating the new algorithms, and provide a detailed discussion about more sophisticated optimization tools that can further be used to solve OT problems thanks to our reformulation.

Owing to their statistical properties, non-convex sparse regularizers have attracted much interest for estimating a sparse linear model from high dimensional data. Given that the solution is sparse, for accelerating convergence, a working set strategy addresses the optimization problem through an iterative algorithm by incre-menting the number of variables to optimize until the identification of the solution support. While those methods have been well-studied and theoretically supported for convex regularizers, this paper proposes a working set algorithm for non-convex sparse regularizers with convergence guarantees. The algorithm, named FireWorks, is based on a non-convex reformulation of a recent primal-dual approach and leverages on the geometry of the residuals. Our theoretical guarantees derive from a lower bound of the objective function decrease between two inner solver iterations and shows the convergence to a stationary point of the full problem. More importantly, we also show that convergence is preserved even when the inner solver is inexact, under sufficient decay of the error across iterations. Our experimental results demonstrate high computational gain when using our working set strategy compared to the full problem solver for both block-coordinate descent or a proximal gradient solver.

We present a 2-step optimal transport approach that performs a mapping from a source distribution to a target distribution. Here, the target has the particularity to present new classes not present in the source domain. The first step of the approach aims at rejecting the samples issued from these new classes using an optimal transport plan. The second step solves the target (class ratio) shift still as an optimal transport problem. We develop a dual approach to solve the optimization problem involved at each step and we prove that our results outperform recent state-of-the-art performances. We further apply the approach to the setting where the source and target distributions present both a label-shift and an increasing covariate (features) shift to show its robustness.

We address the problem of unsupervised domain adaptation under the setting of generalized target shift (both class-conditional and label shifts occur). We show that in that setting, for good generalization, it is necessary to learn with similar source and target label distributions and to match the class-conditional probabilities. For this purpose, we propose an estimation of target label proportion by blending mixture estimation and optimal transport. This estimation comes with theoretical guarantees of correctness. Based on the estimation, we learn a model by minimizing a importance weighted loss and a Wasserstein distance between weighted marginals. We prove that this minimization allows to match class-conditionals given mild assumptions on their geometry. Our experimental results show that our method performs better on average than competitors accross a range domain adaptation problems including digits,VisDA and Office.

Object detection in still images has drawn a lot of attention over past few years, and with the advent of Deep Learning impressive performances have been achieved with numerous industrial applications. Most of these deep learning models rely on RGB images to localize and identify objects in the image. However in some application scenarii, images are compressed either for storage savings or fast transmission. Therefore a time consuming image decompression step is compulsory in order to apply the aforementioned deep models. To alleviate this drawback, we propose a fast deep architecture for object detection in JPEG images, one of the most widespread compression format. We train a neural network to detect objects based on the blockwise DCT (discrete cosine transform) coefficients {issued from} the JPEG compression algorithm. We modify the well-known Single Shot multibox Detector (SSD) by replacing its first layers with one convolutional layer dedicated to process the DCT inputs. Experimental evaluations on PASCAL VOC and industrial dataset comprising images of road traffic surveillance show that the model is about $2\times$ faster than regular SSD with promising detection performances. To the best of our knowledge, this paper is the first to address detection in compressed JPEG images.

Leveraging on the convexity of the Lasso problem , screening rules help in accelerating solvers by discarding irrelevant variables, during the optimization process. However, because they provide better theoretical guarantees in identifying relevant variables, several non-convex regularizers for the Lasso have been proposed in the literature. This work is the first that introduces a screening rule strategy into a non-convex Lasso solver. The approach we propose is based on a iterative majorization-minimization (MM) strategy that includes a screening rule in the inner solver and a condition for propagating screened variables between iterations of MM. In addition to improve efficiency of solvers, we also provide guarantees that the inner solver is able to identify the zeros components of its critical point in finite time. Our experimental analysis illustrates the significant computational gain brought by the new screening rule compared to classical coordinate-descent or proximal gradient descent methods.