919 research outputs found
Kernel Exponential Family Estimation via Doubly Dual Embedding
We investigate penalized maximum log-likelihood estimation for exponential
family distributions whose natural parameter resides in a reproducing kernel
Hilbert space. Key to our approach is a novel technique, doubly dual embedding,
that avoids computation of the partition function. This technique also allows
the development of a flexible sampling strategy that amortizes the cost of
Monte-Carlo sampling in the inference stage. The resulting estimator can be
easily generalized to kernel conditional exponential families. We establish a
connection between kernel exponential family estimation and MMD-GANs, revealing
a new perspective for understanding GANs. Compared to the score matching based
estimators, the proposed method improves both memory and time efficiency while
enjoying stronger statistical properties, such as fully capturing smoothness in
its statistical convergence rate while the score matching estimator appears to
saturate. Finally, we show that the proposed estimator empirically outperforms
state-of-the-artComment: 22 pages, 20 figures; AISTATS 201
Kernel Exponential Family Estimation via Doubly Dual Embedding
We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We establish a connection between kernel exponential family estimation and MMD-GANs, revealing a new perspective for understanding GANs. Compared to the score matching based estimators, the proposed method improves both memory and time efficiency while enjoying stronger statistical properties, such as fully capturing smoothness in its statistical convergence rate while the score matching estimator appears to saturate. Finally, we show that the proposed estimator empirically outperforms state-of-the-art methods in both kernel exponential family estimation and its conditional extension
Exponential Family Estimation via Adversarial Dynamics Embedding
We present an efficient algorithm for maximum likelihood estimation (MLE) of exponential family models, with a general parametrization of the energy function that includes neural networks. We exploit the primal-dual view of the MLE with a kinetics augmented model to obtain an estimate associated with an adversarial dual sampler. To represent this sampler, we introduce a novel neural architecture, dynamics embedding, that generalizes Hamiltonian Monte-Carlo (HMC). The proposed approach inherits the flexibility of HMC while enabling tractable entropy estimation for the augmented model. By learning both a dual sampler and the primal model simultaneously, and sharing parameters between them, we obviate the requirement to design a separate sampling procedure once the model has been trained, leading to more effective learning. We show that many existing estimators, such as contrastive divergence, pseudo/composite-likelihood, score matching, minimum Stein discrepancy estimator, non-local contrastive objectives, noise-contrastive estimation, and minimum probability flow, are special cases of the proposed approach, each expressed by a different (fixed) dual sampler. An empirical investigation shows that adapting the sampler during MLE can significantly improve on state-of-the-art estimators
The -invariant massive Laplacian on isoradial graphs
We introduce a one-parameter family of massive Laplacian operators
defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of , the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at , thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators provide a one-parameter
family of -invariant rooted spanning forest models. When the isoradial graph
is moreover -periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus . We
further show that every Harnack curve of genus with
symmetry arises from such a massive
Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica
Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages
We propose an efficient nonparametric strategy for learning a message
operator in expectation propagation (EP), which takes as input the set of
incoming messages to a factor node, and produces an outgoing message as output.
This learned operator replaces the multivariate integral required in classical
EP, which may not have an analytic expression. We use kernel-based regression,
which is trained on a set of probability distributions representing the
incoming messages, and the associated outgoing messages. The kernel approach
has two main advantages: first, it is fast, as it is implemented using a novel
two-layer random feature representation of the input message distributions;
second, it has principled uncertainty estimates, and can be cheaply updated
online, meaning it can request and incorporate new training data when it
encounters inputs on which it is uncertain. In experiments, our approach is
able to solve learning problems where a single message operator is required for
multiple, substantially different data sets (logistic regression for a variety
of classification problems), where it is essential to accurately assess
uncertainty and to efficiently and robustly update the message operator.Comment: accepted to UAI 2015. Correct typos. Add more content to the
appendix. Main results unchange
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
We present a novel algorithm to estimate the barycenter of arbitrary
probability distributions with respect to the Sinkhorn divergence. Based on a
Frank-Wolfe optimization strategy, our approach proceeds by populating the
support of the barycenter incrementally, without requiring any pre-allocation.
We consider discrete as well as continuous distributions, proving convergence
rates of the proposed algorithm in both settings. Key elements of our analysis
are a new result showing that the Sinkhorn divergence on compact domains has
Lipschitz continuous gradient with respect to the Total Variation and a
characterization of the sample complexity of Sinkhorn potentials. Experiments
validate the effectiveness of our method in practice.Comment: 46 pages, 8 figure
Noise Contrastive Meta-Learning for Conditional Density Estimation using Kernel Mean Embeddings
Current meta-learning approaches focus on learning functional representations
of relationships between variables, i.e. on estimating conditional expectations
in regression. In many applications, however, we are faced with conditional
distributions which cannot be meaningfully summarized using expectation only
(due to e.g. multimodality). Hence, we consider the problem of conditional
density estimation in the meta-learning setting. We introduce a novel technique
for meta-learning which combines neural representation and noise-contrastive
estimation with the established literature of conditional mean embeddings into
reproducing kernel Hilbert spaces. The method is validated on synthetic and
real-world problems, demonstrating the utility of sharing learned
representations across multiple conditional density estimation tasks
- …