CapProNet: Deep Feature Learning via Orthogonal Projections onto Capsule Subspaces

In this paper, we formalize the idea behind capsule nets of using a capsule vector rather than a neuron activation to predict the label of samples. To this end, we propose to learn a group of capsule subspaces onto which an input feature vector is projected. Then the lengths of resultant capsules are used to score the probability of belonging to different classes. We train such a Capsule Projection Network (CapProNet) by learning an orthogonal projection matrix for each capsule subspace, and show that each capsule subspace is updated until it contains input feature vectors corresponding to the associated class. We will also show that the capsule projection can be viewed as normalizing the multiple columns of the weight matrix simultaneously to form an orthogonal basis, which makes it more effective in incorporating novel components of input features to update capsule representations. In other words, the capsule projection can be viewed as a multi-dimensional weight normalization in capsule subspaces, where the conventional weight normalization is simply a special case of the capsule projection onto 1D lines. Only a small negligible computing overhead is incurred to train the network in low-dimensional capsule subspaces or through an alternative hyper-power iteration to estimate the normalization matrix. Experiment results on image datasets show the presented model can greatly improve the performance of the state-of-the-art ResNet backbones by $10-20\%$ and that of the Densenet by $5-7\%$ respectively at the same level of computing and memory expenses. The CapProNet establishes the competitive state-of-the-art performance for the family of capsule nets by significantly reducing test errors on the benchmark datasets.Comment: Liheng Zhang, Marzieh Edraki, Guo-Jun Qi. CapProNet: Deep Feature Learning via Orthogonal Projections onto Capsule Subspaces, in Proccedings of Thirty-second Conference on Neural Information Processing Systems (NIPS 2018), Palais des Congr\`es de Montr\'eal, Montr\'eal, Canda, December 3-8, 201

Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'s are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

A dissipative time reversal technique for photo-acoustic tomography in a cavity

We consider the inverse source problem arising in thermo- and photo-acoustic tomography. It consists in reconstructing the initial pressure from the boundary measurements of the acoustic wave. Our goal is to extend versatile time reversal techniques to the case of perfectly reflecting boundary of the domain. Standard time reversal works only if the solution of the direct problem decays in time, which does not happen in the setup we consider. We thus propose a novel time reversal technique with a non-standard boundary condition. The error induced by this time reversal technique satisfies the wave equation with a dissipative boundary condition and, therefore, decays in time. For larger measurement times, this method yields a close approximation; for smaller times, the first approximation can be iteratively refined, resulting in a convergent Neumann series for the approximation

Gaussian Belief Propagation Based Multiuser Detection

In this work, we present a novel construction for solving the linear multiuser detection problem using the Gaussian Belief Propagation algorithm. Our algorithm yields an efficient, iterative and distributed implementation of the MMSE detector. We compare our algorithm's performance to a recent result and show an improved memory consumption, reduced computation steps and a reduction in the number of sent messages. We prove that recent work by Montanari et al. is an instance of our general algorithm, providing new convergence results for both algorithms.Comment: 6 pages, 1 figures, appeared in the 2008 IEEE International Symposium on Information Theory, Toronto, July 200

Wavelets and Fast Numerical Algorithms

Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see `macros'