We present a probabilistic framework for nonlinearities, based on doubly truncated Gaussian distributions. By setting the truncation points appropriately, we are able to generate various types of nonlinearities within a unified framework, including sigmoid, tanh and ReLU, the most commonly used nonlinearities in neural networks. The framework readily integrates into existing stochastic neural networks (with hidden units characterized as random variables), allowing one for the first time to learn the nonlinearities alongside model weights in these networks. Extensive experiments demonstrate the performance improvements brought about by the proposed framework when integrated with the restricted Boltzmann machine (RBM), temporal RBM and the truncated Gaussian graphical model (TGGM).

Tensor decompositions are invaluable tools in analyzing multimodal datasets. In many real-world scenarios, such datasets are far from being static, to the contrary they tend to grow over time. For instance, in an online social network setting, as we observe new interactions over time, our dataset gets updated in its "time" mode. How can we maintain a valid and accurate tensor decomposition of such a dynamically evolving multimodal dataset, without having to re-compute the entire decomposition after every single update? In this paper we introduce SaMbaTen, a Sampling-based Batch Incremental Tensor Decomposition algorithm, which incrementally maintains the decomposition given new updates to the tensor dataset. SaMbaTen is able to scale to datasets that the state-of-the-art in incremental tensor decomposition is unable to operate on, due to its ability to effectively summarize the existing tensor and the incoming updates, and perform all computations in the reduced summary space. We extensively evaluate SaMbaTen using synthetic and real datasets. Indicatively, SaMbaTen achieves comparable accuracy to state-of-the-art incremental and non-incremental techniques, while being 25-30 times faster. Furthermore, SaMbaTen scales to very large sparse and dense dynamically evolving tensors of dimensions up to 100K x 100K x 100K where state-of-the-art incremental approaches were not able to operate.

The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback-Leibler divergence from the posterior distribution to the approximation, or equivalently maximizing the lower bound for the model evidence. We derive an explicit expression for the lower bound, and show the existence and uniqueness of the optimal Gaussian approximation. The lower bound functional can be viewed as a variant of classical Tikhonov regularization that penalizes also the covariance. Then we develop an efficient alternating direction maximization algorithm for solving the optimization problem, and analyze its convergence. We discuss strategies for reducing the computational complexity via low rank structure of the forward operator and the sparsity of the covariance. Further, as an application of the lower bound, we discuss hierarchical Bayesian modeling for selecting the hyperparameter in the prior distribution, and propose a monotonically convergent algorithm for determining the hyperparameter. We present extensive numerical experiments to illustrate the Gaussian approximation and the algorithms.

Identifying context-specific entity networks from aggregated data is an important task, arising often in bioinformatics and neuroimaging. Computationally, this task can be formulated as jointly estimating multiple different, but related, sparse Undirected Graphical Models (UGM) from aggregated samples across several contexts. Previous joint-UGM studies have mostly focused on sparse Gaussian Graphical Models (sGGMs) and can't identify context-specific edge patterns directly. We, therefore, propose a novel approach, SIMULE (detecting Shared and Individual parts of MULtiple graphs Explicitly) to learn multi-UGM via a constrained L1 minimization. SIMULE automatically infers both specific edge patterns that are unique to each context and shared interactions preserved among all the contexts. Through the L1 constrained formulation, this problem is cast as multiple independent subtasks of linear programming that can be solved efficiently in parallel. In addition to Gaussian data, SIMULE can also handle multivariate Nonparanormal data that greatly relaxes the normality assumption that many real-world applications do not follow. We provide a novel theoretical proof showing that SIMULE achieves a consistent result at the rate O(log(Kp)/n_{tot}). On multiple synthetic datasets and two biomedical datasets, SIMULE shows significant improvement over state-of-the-art multi-sGGM and single-UGM baselines.

Did you know using XGBoost algorithm is one of the popular winning recipe of data science competitions? So, what makes it more powerful than a traditional Random Forest or Neural Network? In the last few years, predictive modeling has become much faster and accurate. I remember spending long hours on feature engineering for improving model by few decimals. A lot of that difficult work, can now be done by using better algorithms.

Another TensorFlow feature you typically want to use is checkpointing – saving the parameters of your model to restore them later on. Checkpoints can be used to continue training at a later point, or to pick the best parameters setting using early stopping. Checkpoints are created using a Saver object. Before we can train our model we also need to initialize the variables in our graph. The initialize_all_variables function is a convenience function run all of the initializers we've defined for our variables. You can also call the initializer of your variables manually. That's useful if you want to initialize your embeddings with pre-trained values for example. Let's now define a function for a single training step, evaluating the model on a batch of data and updating the model parameters.

Self-driving cars, face detection software, and voice controlled speakers all are built on machine learning technologies and frameworks--and these are just the first wave. Over the next decade, a new generation of products will transform our world, initiating new approaches to software development and the applications and products that we create and use. As a Java developer, you want to get ahead of this curve now--when tech companies are beginning to seriously invest in machine learning. What you learn today, you can build on over the next five years, but you have to start somewhere. This article will get you started.

Two popular univariate time series methods are Exponential Smoothing (e.g., Holt-Winters) and ARIMA(Autoregressive Integrated Moving Average). Causal variables will typically include data such as GRPs and price and also may incorporate data from consumer surveys or exogenous variables such as GDP. Vector Autoregression (VAR), the Vector Error Correction Model (VECM) and the more general State Space framework are three frequently-used approaches to multiple time series analysis. Causal data can be included and Market Response/Marketing Mix modeling conducted.

We use different equations depending on the number of output classes. With 2 classes, we will use binomial cross entropy for loss and more than 2 classes involves using the cross-entropy with softmax. Example of how to calculate the cross-entropy loss for a 3 class problem. With the multinomial cross entropy, you can see that we only keep the loss contribution from the correct class. Usually, with neural nets, this will be case if our ouputs are sparse (just 1 true class).

Principal component analysis (PCA) is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Only few works have proposed a probabilistic approach able to infer the number of significant components. To this purpose, this paper introduces a Bayesian nonparametric principal component analysis (BNP-PCA). The proposed model projects observations onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. The prior on factor scores involves an Indian buffet process to model the uncertainty related to the number of components. The parameters of interest as well as the nuisance parameters are finally inferred within a fully Bayesian framework via Monte Carlo sampling. A study of the (in-)consistence of the marginal maximum a posteriori estimator of the latent dimension is carried out. A new estimator of the subspace dimension is proposed. Moreover, for sake of statistical significance, a Kolmogorov-Smirnov test based on the posterior distribution of the principal components is used to refine this estimate. The behaviour of the algorithm is first studied on various synthetic examples. Finally, the proposed BNP dimension reduction approach is shown to be easily yet efficiently coupled with clustering or latent factor models within a unique framework.