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Register a new userCo-clustering through Optimal Transport
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Added by
Charlotte Laclau
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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In this paper, we present a novel method for co-clustering, an unsupervised learning approach that aims at discovering homogeneous groups of data instances and features by grouping them simultaneously. The proposed method uses the entropy regularized optimal transport between empirical measures defined on data instances and features in order to obtain an estimated joint probability density function represented by the optimal coupling matrix. This matrix is further factorized to obtain the induced row and columns partitions using multiscale representations approach. To justify our method theoretically, we show how the solution of the regularized optimal transport can be seen from the variational inference perspective thus motivating its use for co-clustering. The algorithm derived for the proposed method and its kernelized version based on the notion of Gromov-Wasserstein distance are fast, accurate and can determine automatically the number of both row and column clusters. These features are vividly demonstrated through extensive experimental evaluations.
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We study multi-marginal optimal transport, a generalization of optimal transport that allows us to define discrepancies between multiple measures. It provides a framework to solve multi-task learning problems and to perform barycentric averaging. However, multi-marginal distances between multiple measures are typically challenging to compute because they require estimating a transport plan with $N^P$ variables. In this paper, we address this issue in the following way: 1) we efficiently solve the one-dimensional multi-marginal Monge-Wasserstein problem for a classical cost function in closed form, and 2) we propose a higher-dimensional multi-marginal discrepancy via slicing and study its generalized metric properties. We show that computing the sliced multi-marginal discrepancy is massively scalable for a large number of probability measures with support as large as $10^7$ samples. Our approach can be applied to solving problems such as barycentric averaging, multi-task density estimation and multi-task reinforcement learning.
Missing data is a crucial issue when applying machine learning algorithms to real-world datasets. Starting from the simple assumption that two batches extracted randomly from the same dataset should share the same distribution, we leverage optimal transport distances to quantify that criterion and turn it into a loss function to impute missing data values. We propose practical methods to minimize these losses using end-to-end learning, that can exploit or not parametric assumptions on the underlying distributions of values. We evaluate our methods on datasets from the UCI repository, in MCAR, MAR and MNAR settings. These experiments show that OT-based methods match or out-perform state-of-the-art imputation methods, even for high percentages of missing values.
The recent advances in single-cell technologies have enabled us to profile genomic features at unprecedented resolution and datasets from multiple domains are available, including datasets that profile different types of genomic features and datasets that profile the same type of genomic features across different species. These datasets typically have different powers in identifying the unknown cell types through clustering, and data integration can potentially lead to a better performance of clustering algorithms. In this work, we formulate the problem in an unsupervised transfer learning framework, which utilizes knowledge learned from auxiliary dataset to improve the clustering performance of target dataset. The degree of shared information among the target and auxiliary datasets can vary, and their distributions can also be different. To address these challenges, we propose an elastic coupled co-clustering based transfer learning algorithm, by elastically propagating clustering knowledge obtained from the auxiliary dataset to the target dataset. Implementation on single-cell genomic datasets shows that our algorithm greatly improves clustering performance over the traditional learning algorithms. The source code and data sets are available at https://github.com/cuhklinlab/elasticC3.
In many machine learning applications, it is necessary to meaningfully aggregate, through alignment, different but related datasets. Optimal transport (OT)-based approaches pose alignment as a divergence minimization problem: the aim is to transform a source dataset to match a target dataset using the Wasserstein distance as a divergence measure. We introduce a hierarchical formulation of OT which leverages clustered structure in data to improve alignment in noisy, ambiguous, or multimodal settings. To solve this numerically, we propose a distributed ADMM algorithm that also exploits the Sinkhorn distance, thus it has an efficient computational complexity that scales quadratically with the size of the largest cluster. When the transformation between two datasets is unitary, we provide performance guarantees that describe when and how well aligned cluster correspondences can be recovered with our formulation, as well as provide worst-case dataset geometry for such a strategy. We apply this method to synthetic datasets that model data as mixtures of low-rank Gaussians and study the impact that different geometric properties of the data have on alignment. Next, we applied our approach to a neural decoding application where the goal is to predict movement directions and instantaneous velocities from populations of neurons in the macaque primary motor cortex. Our results demonstrate that when clustered structure exists in datasets, and is consistent across trials or time points, a hierarchical alignment strategy that leverages such structure can provide significant improvements in cross-domain alignment.
Optimal transport aims to estimate a transportation plan that minimizes a displacement cost. This is realized by optimizing the scalar product between the sought plan and the given cost, over the space of doubly stochastic matrices. When the entropy regularization is added to the problem, the transportation plan can be efficiently computed with the Sinkhorn algorithm. Thanks to this breakthrough, optimal transport has been progressively extended to machine learning and statistical inference by introducing additional application-specific terms in the problem formulation. It is however challenging to design efficient optimization algorithms for optimal transport based extensions. To overcome this limitation, we devise a general forward-backward splitting algorithm based on Bregman distances for solving a wide range of optimization problems involving a differentiable function with Lipschitz-continuous gradient and a doubly stochastic constraint. We illustrate the efficiency of our approach in the context of continuous domain adaptation. Experiments show that the proposed method leads to a significant improvement in terms of speed and performance with respect to the state of the art for domain adaptation on a continually rotating distribution coming from the standard two moon dataset.
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