A Fast Multi-Layer Approximation to Semi-Discrete Optimal Transport

International audienceThe optimal transport (OT) framework has been largely used in inverse imaging and computer vision problems, as an interesting way to incorporate statistical constraints or priors. In recent years, OT has also been used in machine learning, mostly as a metric to compare probability distributions. This work addresses the semi-discrete OT problem where a continuous source distribution is matched to a discrete target distribution. We introduce a fast stochastic algorithm to approximate such a semi-discrete OT problem using a hierarchical multi-layer transport plan. This method allows for tractable computation in high-dimensional case and for large point-clouds, both during training and synthesis time. Experiments demonstrate its numerical advantage over multi-scale (or multi-level) methods. Applications to fast exemplar-based texture synthesis based on patch matching with two layers, also show stunning improvements over previous single layer approaches. This shallow model achieves comparable results with state-of-the-art deep learning methods, while being very compact, faster to train, and using a single image during training instead of a large dataset

Enabling Quality-Driven Scalable Video Transmission over Multi-User NOMA System

Recently, non-orthogonal multiple access (NOMA) has been proposed to achieve higher spectral efficiency over conventional orthogonal multiple access. Although it has the potential to meet increasing demands of video services, it is still challenging to provide high performance video streaming. In this research, we investigate, for the first time, a multi-user NOMA system design for video transmission. Various NOMA systems have been proposed for data transmission in terms of throughput or reliability. However, the perceived quality, or the quality-of-experience of users, is more critical for video transmission. Based on this observation, we design a quality-driven scalable video transmission framework with cross-layer support for multi-user NOMA. To enable low complexity multi-user NOMA operations, a novel user grouping strategy is proposed. The key features in the proposed framework include the integration of the quality model for encoded video with the physical layer model for NOMA transmission, and the formulation of multi-user NOMA-based video transmission as a quality-driven power allocation problem. As the problem is non-concave, a global optimal algorithm based on the hidden monotonic property and a suboptimal algorithm with polynomial time complexity are developed. Simulation results show that the proposed multi-user NOMA system outperforms existing schemes in various video delivery scenarios.Comment: 9 pages, 6 figures. This paper has already been accepted by IEEE INFOCOM 201

A numerical algorithm for L2L_2 semi-discrete optimal transport in 3D

This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh), and where $\nu$ is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure $\mu$. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.Comment: 23 pages, 14 figure

Optimal Transport for Domain Adaptation

Domain adaptation from one data space (or domain) to another is one of the most challenging tasks of modern data analytics. If the adaptation is done correctly, models built on a specific data space become more robust when confronted to data depicting the same semantic concepts (the classes), but observed by another observation system with its own specificities. Among the many strategies proposed to adapt a domain to another, finding a common representation has shown excellent properties: by finding a common representation for both domains, a single classifier can be effective in both and use labelled samples from the source domain to predict the unlabelled samples of the target domain. In this paper, we propose a regularized unsupervised optimal transportation model to perform the alignment of the representations in the source and target domains. We learn a transportation plan matching both PDFs, which constrains labelled samples in the source domain to remain close during transport. This way, we exploit at the same time the few labeled information in the source and the unlabelled distributions observed in both domains. Experiments in toy and challenging real visual adaptation examples show the interest of the method, that consistently outperforms state of the art approaches