In this section, we will elaborate on the derivation and the properties of the discrete Fourier transform. A.1 From Fourier transform to discrete Fourier transform Discrete Fourier transform (DFT) can be derived in many ways. To name a few basic ones, the FT of a unit impulse function (a.k.a. However, we rarely deal with continuous signal in the real application. The extension from 1D DFT to 2D DFT is straightforward.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP), which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts. We test the resulting method, which we call Continuous Orthogonal Matching Pursuit (COMP), on simulated and neural data, where it shows gains over CBP in both speed and accuracy.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP), which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts.

The majority of signal data captured in the real world uses numerous sensors with different resolutions. In practice, however, most deep learning architectures are fixed-resolution; they consider a single resolution at training time and inference time. This is convenient to implement but fails to fully take advantage of the diverse signal data that exists. In contrast, other deep learning architectures are adaptive-resolution; they directly allow various resolutions to be processed at training time and inference time. This benefits robustness and computational efficiency but introduces difficult design constraints that hinder mainstream use. In this work, we address the shortcomings of both fixed-resolution and adaptive-resolution methods by introducing Adaptive Resolution Residual Networks (ARRNs), which inherit the advantages of adaptive-resolution methods and the ease of use of fixed-resolution methods. We construct ARRNs from Laplacian residuals, which serve as generic adaptive-resolution adapters for fixed-resolution layers, and which allow casting high-resolution ARRNs into low-resolution ARRNs at inference time by simply omitting high-resolution Laplacian residuals, thus reducing computational cost on low-resolution signals without compromising performance. We complement this novel component with Laplacian dropout, which regularizes for robustness to a distribution of lower resolutions, and which also regularizes for errors that may be induced by approximate smoothing kernels in Laplacian residuals. We provide a solid grounding for the advantageous properties of ARRNs through a theoretical analysis based on neural operators, and empirically show that ARRNs embrace the challenge posed by diverse resolutions with greater flexibility, robustness, and computational efficiency.

This interactive paper aims to provide an intuitive understanding of the self-calibrating interface paradigm. Under this paradigm, you can choose how to use an interface which can adapt to your preferences on the fly. We introduce a PIN entering task and gradually release constraints, moving from a pre-calibrated interface to a self-calibrating interface while increasing the complexity of input modalities from buttons, to points on a map, to sketches, and finally to spoken words. This is not a traditional research paper with a hypothesis and experimental results to support claims; the research supporting this work has already been done and we refer to it extensively in the later sections. Instead, our aim is to walk you through an intriguing interaction paradigm in small logical steps with supporting illustrations, interactive demonstrations, and videos to reinforce your learning. We designed this paper for the enjoyments of curious minds of any backgrounds, it is written in plain English and no prior knowledge is necessary. All demos are available online at openvault.jgrizou.com

Machine learning (ML) projects have become a leading tool in enormous domains of the computer industry. Their rule is far beyond computational aspects. Indeed, they are a focal point in designing analytical business decisions. The commercial usage of these models raises new challenges. The ML academic research often assumes that: The data in the database represents well the global data distribution.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP), which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts.