Deep structured models are widely used for tasks like semantic segmentation, where explicit correlations between variables provide important prior information which generally helps to reduce the data needs of deep nets. However, current deep structured models are restricted by oftentimes very local neighborhood structure, which cannot be increased for computational complexity reasons, and by the fact that the output configuration, or a representation thereof, cannot be transformed further. Very recent approaches which address those issues include graphical model inference inside deep nets so as to permit subsequent non-linear output space transformations. However, optimization of those formulations is challenging and not well understood. Here, we develop a novel model which generalizes existing approaches, such as structured prediction energy networks, and discuss a formulation which maintains applicability of existing inference techniques.
Statistically independent features can be extracted by finding a factorial representationof a signal distribution. Principal Component Analysis (PCA) accomplishes this for linear correlated and Gaussian distributedsignals. Independent Component Analysis (ICA), formalized by Comon (1994), extracts features in the case of linear statisticaldependent but not necessarily Gaussian distributed signals. Nonlinear Component Analysis finally should find a factorial representationfor nonlinear statistical dependent distributed signals. This paper proposes for this task a novel feed-forward, information conserving, nonlinear map - the explicit symplectic transformations. It also solves the problem of non-Gaussian output distributions by considering single coordinate higher order statistics. 1 Introduction In previous papers Deco and Brauer (1994) and Parra, Deco, and Miesbach (1995) suggest volume conserving transformations and factorization as the key elements for a nonlinear version of Independent Component Analysis. As a general class of volume conserving transformations Parra et al. (1995) propose the symplectic transformation. It was defined by an implicit nonlinear equation, which leads to a complex relaxation procedure for the function recall. In this paper an explicit form of the symplectic map is proposed, overcoming thus the computational problems.
Here we are, part four of our Practicing Deep Learning Series. Alas, we are at the point where we will start to examine multilayer neural networks! We've spent a decent amount of time building up to this point, but with good reason. It can be easy to gloss over the details of the individual neurons and we feel the risk of being too verbose outweighed the risk of being uninformative. The real power of neural networks become increasingly apparent when we start making multi-layer networks. In this post, we're going to describe the basic multi-layer network and look at examples of some of simple tasks it can solve. We're going to dive into two specific examples, but we provide code for those two – plus a few others. We'll point you to that code as we go.
A neural network learning paradigm based on information theory is proposed asa way to perform in an unsupervised fashion, redundancy reduction among the elements of the output layer without loss of information fromthe sensory input. The model developed performs nonlinear decorrelation up to higher orders of the cumulant tensors and results in probabilistically independent components of the output layer. This means that we don't need to assume Gaussian distribution neither at the input nor at the output. The theory presented is related to the unsupervised-learning theoryof Barlow, which proposes redundancy reduction as the goal of cognition. When nonlinear units are used nonlinear principal componentanalysis is obtained.
Less work has been performed on using neural networks to process floating point numbers and it is sometimes stated that neural networks are somehow inherently inaccurate andtherefore best suited for "fuzzy" qualitative reasoning. Nevertheless, the potential speed of massively parallel operations make neural net "number crunching" an interesting topic to explore. In this paper we discuss some of our work in which we demonstrate that for certain applications neural networks can achieve significantly higher numerical accuracy than more conventional techniques. Inparticular, prediction of future values of a chaotic time series can be performed with exceptionally high accuracy. We analyze how a neural net is able to do this, and in the process show that a large class of functions from Rn. Rffl may be accurately approximated by a backpropagation neural net with just two "hidden" layers. The network uses this functional approximation to perform either interpolation (signal processing applications) or extrapolation (symbol processing applicationsJ. Neural nets therefore use quite familiar methods toperform.