324 research outputs found
Minimax estimation of smooth optimal transport maps
Brenier's theorem is a cornerstone of optimal transport that guarantees the
existence of an optimal transport map between two probability distributions
and over under certain regularity conditions. The main
goal of this work is to establish the minimax estimation rates for such a
transport map from data sampled from and under additional smoothness
assumptions on . To achieve this goal, we develop an estimator based on the
minimization of an empirical version of the semi-dual optimal transport
problem, restricted to truncated wavelet expansions. This estimator is shown to
achieve near minimax optimality using new stability arguments for the semi-dual
and a complementary minimax lower bound. Furthermore, we provide numerical
experiments on synthetic data supporting our theoretical findings and
highlighting the practical benefits of smoothness regularization. These are the
first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure
Learning Generative Models across Incomparable Spaces
Generative Adversarial Networks have shown remarkable success in learning a
distribution that faithfully recovers a reference distribution in its entirety.
However, in some cases, we may want to only learn some aspects (e.g., cluster
or manifold structure), while modifying others (e.g., style, orientation or
dimension). In this work, we propose an approach to learn generative models
across such incomparable spaces, and demonstrate how to steer the learned
distribution towards target properties. A key component of our model is the
Gromov-Wasserstein distance, a notion of discrepancy that compares
distributions relationally rather than absolutely. While this framework
subsumes current generative models in identically reproducing distributions,
its inherent flexibility allows application to tasks in manifold learning,
relational learning and cross-domain learning.Comment: International Conference on Machine Learning (ICML
A review of domain adaptation without target labels
Domain adaptation has become a prominent problem setting in machine learning
and related fields. This review asks the question: how can a classifier learn
from a source domain and generalize to a target domain? We present a
categorization of approaches, divided into, what we refer to as, sample-based,
feature-based and inference-based methods. Sample-based methods focus on
weighting individual observations during training based on their importance to
the target domain. Feature-based methods revolve around on mapping, projecting
and representing features such that a source classifier performs well on the
target domain and inference-based methods incorporate adaptation into the
parameter estimation procedure, for instance through constraints on the
optimization procedure. Additionally, we review a number of conditions that
allow for formulating bounds on the cross-domain generalization error. Our
categorization highlights recurring ideas and raises questions important to
further research.Comment: 20 pages, 5 figure
Generative Adversarial Networks (GANs): Challenges, Solutions, and Future Directions
Generative Adversarial Networks (GANs) is a novel class of deep generative
models which has recently gained significant attention. GANs learns complex and
high-dimensional distributions implicitly over images, audio, and data.
However, there exists major challenges in training of GANs, i.e., mode
collapse, non-convergence and instability, due to inappropriate design of
network architecture, use of objective function and selection of optimization
algorithm. Recently, to address these challenges, several solutions for better
design and optimization of GANs have been investigated based on techniques of
re-engineered network architectures, new objective functions and alternative
optimization algorithms. To the best of our knowledge, there is no existing
survey that has particularly focused on broad and systematic developments of
these solutions. In this study, we perform a comprehensive survey of the
advancements in GANs design and optimization solutions proposed to handle GANs
challenges. We first identify key research issues within each design and
optimization technique and then propose a new taxonomy to structure solutions
by key research issues. In accordance with the taxonomy, we provide a detailed
discussion on different GANs variants proposed within each solution and their
relationships. Finally, based on the insights gained, we present the promising
research directions in this rapidly growing field.Comment: 42 pages, Figure 13, Table
A Principled Approach for Learning Task Similarity in Multitask Learning
Multitask learning aims at solving a set of related tasks simultaneously, by
exploiting the shared knowledge for improving the performance on individual
tasks. Hence, an important aspect of multitask learning is to understand the
similarities within a set of tasks. Previous works have incorporated this
similarity information explicitly (e.g., weighted loss for each task) or
implicitly (e.g., adversarial loss for feature adaptation), for achieving good
empirical performances. However, the theoretical motivations for adding task
similarity knowledge are often missing or incomplete. In this paper, we give a
different perspective from a theoretical point of view to understand this
practice. We first provide an upper bound on the generalization error of
multitask learning, showing the benefit of explicit and implicit task
similarity knowledge. We systematically derive the bounds based on two distinct
task similarity metrics: H divergence and Wasserstein distance. From these
theoretical results, we revisit the Adversarial Multi-task Neural Network,
proposing a new training algorithm to learn the task relation coefficients and
neural network parameters iteratively. We assess our new algorithm empirically
on several benchmarks, showing not only that we find interesting and robust
task relations, but that the proposed approach outperforms the baselines,
reaffirming the benefits of theoretical insight in algorithm design
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