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The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets…
Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct…
This paper presents a new family of localized orthonormal bases - sinlets - which are well suited for both signal and image processing and analysis. One-dimensional sinlets are related to specific solutions of the time-dependent harmonic…
The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors, matrices, etc) into an orthonormal basis (a set of orthogonal, unit-length vectors, bi or tri dimensional matrices). The…
Generative Adversarial Networks (GANs) advance face synthesis through learning the underlying distribution of observed data. Despite the high-quality generated faces, some minority groups can be rarely generated from the trained models due…
Recent advances in face super-resolution research have utilized the Transformer architecture. This method processes the input image into a series of small patches. However, because of the strong correlation between different facial…
Multiresolution analysis (MRA) on compact abelian group $G$ has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on $L^p(G)$, $1\le p<\infty$. With the help of this scaling…
This paper exploits the multiresolution analysis in the fault analysis on transmission lines. Faults were simulated using the ATP (Alternative Transient Program), considering signals at 128/cycle. A nonorthogonal multiresolution analysis…
Our previous multiscale graph basis dictionaries/graph signal transforms -- Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives -- were…
Detection of interacting and conversational groups from images has applications in video surveillance and social robotics. In this paper we build on prior attempts to find conversational groups by detection of social gathering spaces called…
Group convolutional neural networks are a useful tool for utilizing symmetries known to be in a signal; however, they require that the signal is defined on the group itself. Existing approaches either work directly with group signals, or…
We propose a multiresolution Gaussian process to capture long-range, non-Markovian dependencies while allowing for abrupt changes. The multiresolution GP hierarchically couples a collection of smooth GPs, each defined over an element of a…
Exploration of bias has significant impact on the transparency and applicability of deep learning pipelines in medical settings, yet is so far woefully understudied. In this paper, we consider two separate groups for which training data is…
We propose a new algorithm for training generative adversarial networks that jointly learns latent codes for both identities (e.g. individual humans) and observations (e.g. specific photographs). By fixing the identity portion of the latent…
The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical…
Orthogonal wavelet transforms are a cornerstone of modern signal and image denoising because they combine multiscale representation, energy preservation, and perfect reconstruction. In this paper, we show that these advantages can be…
The problem of detecting and quantifying the presence of symmetries in datasets is useful for model selection, generative modeling, and data analysis, amongst others. While existing methods for hard-coding transformations in neural networks…
Spectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions…
Transformer structure, stacked by a sequence of encoder and decoder network layers, achieves significant development in neural machine translation. However, vanilla Transformer mainly exploits the top-layer representation, assuming the…
We introduce group crosscoders, an extension of crosscoders that systematically discover and analyse symmetrical features in neural networks. While neural networks often develop equivariant representations without explicit architectural…