Deep neural networks have achieved great successes on various machine learning tasks, however, there are many open fundamental questions to be answered. In this paper, we tackle the problem of quantifying the quality of learned wights of different networks with possibly different architectures, going beyond considering the final classification error as the only metric. We introduce \emphExtended Data Jacobian Matrix to help analyze properties of networks of various structures, finding that, the spectrum of the extended data jacobian matrix is a strong discriminating factor for networks of different structures and performance. Based on such observation, we propose a novel regularization method, which manages to improve the network performance comparably to dropout, which in turn verifies the observation.

Latent variable prediction models, such as multi-layer networks, impose auxiliary latent variables between inputs and outputs to allow automatic inference of implicit features useful for prediction. Unfortunately, such models are difficult to train because inference over latent variables must be performed concurrently with parameter optimization---creating a highly non-convex problem. Instead of proposing another local training method, we develop a convex relaxation of hidden-layer conditional models that admits global training. Our approach extends current convex modeling approaches to handle two nested nonlinearities separated by a non-trivial adaptive latent layer. The resulting methods are able to acquire two-layer models that cannot be represented by any single-layer model over the same features, while improving training quality over local heuristics.

Despite the variety of robust regression methods that have been developed, current regression formulations are either NP-hard, or allow unbounded response to even a single leverage point. We present a general formulation for robust regression --Variational M-estimation--that unifies a number of robust regression methods while allowing a tractable approximation strategy. We develop an estimator that requires only polynomial-time, while achieving certain robustness and consistency guarantees. An experimental evaluation demonstrates the effectiveness of the new estimation approach compared to standard methods.