# Non-Parametric Model

http://www.math.wm.edu/~leemis/2008amstat.pdf Univariate Distribution Relationships http://www.math.wm.edu/~leemis/chart/UDR/UDR.html

https://arxiv.org/abs/1610.08936 Learning Scalable Deep Kernels with Recurrent Structure

To model such structure, we propose expressive closed-form kernel functions for Gaussian processes. The resulting model, GP-LSTM, fully encapsulates the inductive biases of long short-term memory (LSTM) recurrent networks, while retaining the non-parametric probabilistic advantages of Gaussian processes.

http://arxiv.org/pdf/1607.05690v1.pdf Stochastic Backpropagation through Mixture Density Distributions

The ability to backpropagate stochastic gradients through continuous latent distributions has been crucial to the emergence of variational autoencoders. The key ingredient is an unbiased and low-variance way of estimating gradients with respect to distribution parameters from gradients evaluated at distribution samples. The “reparameterization trick” [6] provides a class of transforms yielding such estimators for many continuous distributions, including the Gaussian and other members of the location-scale family. However the trick does not readily extend to mixture density models, due to the difficulty of reparameterizing the discrete distribution over mixture weights.

https://www.lix.polytechnique.fr/~nielsen/FN-SIMBAD2013.pdf

http://openreview.net/pdf?id=BJK3Xasel Non-Parametric Neural Network

http://openreview.net/pdf?id=S1jE5L5gl THE CONCRETE DISTRIBUTION: A CONTINUOUS RELAXATION OF DISCRETE RANDOM VARIABLES

While many continuous random variables have such reparameterizations, discrete random variables lack continuous reparameterizations due to the discontinuous nature of discrete states. In this work we introduce concrete random variables – continuous relaxations of discrete random variables. The concrete distribution is a new family of distributions with closed form densities and a simple reparameterization. Whenever a discrete stochastic node of a computation graph can be refactored into a one-hot bit representation that is treated continuously, concrete stochastic nodes can be used with automatic differentiation to produce low-variance biased gradients of objectives (including objectives that depend on the log-probability of latent stochastic nodes) on the corresponding discrete graph. We demonstrate effectiveness of concrete relaxations on density estimation and structured prediction tasks using neural networks.

https://arxiv.org/abs/1611.01144v2 Categorical Reparameterization with Gumbel-Softmax

Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification.

https://arxiv.org/abs/1611.00336 Stochastic Variational Deep Kernel Learning

We propose a novel deep kernel learning model and stochastic variational inference procedure which generalizes deep kernel learning approaches to enable classification, multi-task learning, additive covariance structures, and stochastic gradient training. Specifically, we apply additive base kernels to subsets of output features from deep neural architectures, and jointly learn the parameters of the base kernels and deep network through a Gaussian process marginal likelihood objective. Within this framework, we derive an efficient form of stochastic variational inference which leverages local kernel interpolation, inducing points, and structure exploiting algebra. We show improved performance over stand alone deep networks, SVMs, and state of the art scalable Gaussian processes on several classification benchmarks, including an airline delay dataset containing 6 million training points, CIFAR, and ImageNet.

https://arxiv.org/abs/1612.00377v1 Multi-modal Variational Encoder-Decoders

Recent advances in neural variational inference have facilitated efficient training of powerful directed graphical models with continuous latent variables, such as variational autoencoders. However, these models usually assume simple, uni-modal priors - such as the multivariate Gaussian distribution - yet many real-world data distributions are highly complex and multi-modal. Examples of complex and multi-modal distributions range from topics in newswire text to conversational dialogue responses. When such latent variable models are applied to these domains, the restriction of the simple, uni-modal prior hinders the overall expressivity of the learned model as it cannot possibly capture more complex aspects of the data distribution. To overcome this critical restriction, we propose a flexible, simple prior distribution which can be learned efficiently and potentially capture an exponential number of modes of a target distribution. We develop the multi-modal variational encoder-decoder framework and investigate the effectiveness of the proposed prior in several natural language processing modeling tasks, including document modeling and dialogue modeling.

https://arxiv.org/abs/1706.04161 Lost Relatives of the Gumbel Trick

The Gumbel trick is a method to sample from a discrete probability distribution, or to estimate its normalizing partition function. The method relies on repeatedly applying a random perturbation to the distribution in a particular way, each time solving for the most likely configuration. We derive an entire family of related methods, of which the Gumbel trick is one member, and show that the new methods have superior properties in several settings with minimal additional computational cost. In particular, for the Gumbel trick to yield computational benefits for discrete graphical models, Gumbel perturbations on all configurations are typically replaced with so-called low-rank perturbations. We show how a subfamily of our new methods adapts to this setting, proving new upper and lower bounds on the log partition function and deriving a family of sequential samplers for the Gibbs distribution. Finally, we balance the discussion by showing how the simpler analytical form of the Gumbel trick enables additional theoretical results.

https://arxiv.org/abs/1712.05440v1 Nonparametric Neural Networks

We introduce *nonparametric neural networks*, a non-probabilistic framework for conducting optimization over all possible network sizes and prove its soundness when network growth is limited via an L_p penalty. We train networks under this framework by continuously adding new units while eliminating redundant units via an L_2 penalty. We employ a novel optimization algorithm, which we term *adaptive radial-angular gradient descent* or *AdaRad*, and obtain promising results.

https://arxiv.org/abs/1803.03764 Variance Networks: When Expectation Does Not Meet Your Expectations

Variance networks represent a diverse ensemble that is more robust to adversarial attacks than conventional low-variance ensembles. The success of this model raises several counter-intuitive implications for the training and application of Deep Learning models.

https://arxiv.org/abs/1806.04326v1 Differentiable Compositional Kernel Learning for Gaussian Processes

https://arxiv.org/abs/1805.08498v2 Implicit Reparameterization Gradients

We introduce an alternative approach to computing reparameterization gradients based on implicit differentiation and demonstrate its broader applicability by applying it to Gamma, Beta, Dirichlet, and von Mises distributions, which cannot be used with the classic reparameterization trick. Our experiments show that the proposed approach is faster and more accurate than the existing gradient estimators for these distributions.

https://arxiv.org/abs/1807.01622 Neural Processes

https://arxiv.org/abs/1807.01613 Conditional Neural Processes

https://arxiv.org/pdf/1808.10805.pdf Spherical Latent Spaces for Stable Variational Autoencoders

An analysis of the properties of our vMF representations shows that they learn richer and more nuanced structures in their latent representations than their Gaussian counterparts.

https://arxiv.org/abs/1810.09583 Model Selection Techniques – An Overview