arXiv subjects

Nguyen T. Thao

Publications and source records attributed to Nguyen T. Thao.

8 recordsLinked to original sources

arXiv2026

A residual-iteration framework for alternating projections between affine subspaces

We reformulate the problem of alternating projections between two affine subspaces of a Hilbert space as the minimization of a least-squares functional associated with a bounded linear operator. This viewpoint reveals that classical alternating projections coincide with the unit-step Landweber iteration and enables the introduction of a general residual-state iteration framework that encompasses Landweber, its steepest-descent variant, and the conjugate-gradient method. Within this framework, we establish abstract convergence principles based on residual extinction and translation equivariance, allowing convergence analyses to be carried out once at the level of least-squares optimization and then transferred directly to alternating projection algorithms. As applications, we obtain new variants of alternating projections accelerated by steepest descent and conjugate gradients, together with convergence guarantees in both the consistent and inconsistent settings. We also establish linear convergence results under closed-range assumptions and express the convergence rates explicitly in terms of the Friedrichs angle and the largest principal angle between the underlying subspaces.

math.FA↗

arXiv2025

Bandlimited signal reconstruction from orthogonally-projected data

We show that a broad class of signal acquisition schemes can be interpreted as recording data from a signal $x$ in a space $\cal U$ (typically, though not exclusively, a space of bandlimited functions) via an orthogonal projection $w = P_{\cal V} x$ onto another space $\cal V$. A basic reconstruction method in this case consists in alternating projections between the input space $\cal U$ and the affine space $\cal W$ of signals $u$ satisfying $P_{\cal V} u = w$ (POCS method). Although this method is classically known to be slow, our work reveals new insights and contributions: (i) it applies to new complex encoders emerging from event-based sampling, for which no faster reconstruction method is currently available; (ii) beyond perfect reconstruction, it converges robustly under insufficient (e.g., sub-Nyquist) or inconsistent data (due to noise or errors); (iii) the limit of convergence achieves optimal least-squares approximations under such conditions; (iv) semi-convergence inherently results in regularized reconstructions under ill-posed data acquisition conditions; (v) when $w$ is produced by discrete sampling, the iterative method can be rigorously discretized for DSP implementation -- even with non-separable input spaces $\cal U$. While moving beyond the traditional focus on perfect reconstruction in harmonic analysis, our analysis preserves the deterministic framework of infinite-dimensional Hilbert spaces, consistent with Shannon's sampling theory. For illustration, we apply our proposed theory to two contrasting sampling situations: multi-channel time encoding and nonuniform point sampling in a Sobolev space.

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arXiv2024

Pseudo-inverse reconstruction of bandlimited signals from nonuniform generalized samples with orthogonal kernels

Contrary to the traditional pursuit of research on nonuniform sampling of bandlimited signals, the objective of the present paper is not to find sampling conditions that permit perfect reconstruction, but to perform the best possible signal recovery from any given set of nonuniform samples, whether it is finite as in practice, or infinite to achieve the possibility of unique reconstruction in $L^2({\mathbb R})$. This leads us to consider the pseudo-inverse of the whole sampling map as a linear operator of Hilbert spaces. We propose in this paper an iterative algorithm that systematically performs this pseudo-inversion under the following conditions: (i) the input lies in some closed space $\mathscr A$ (such as a space of bandlimited functions); (ii) the samples are formed by inner product of the input with given kernel functions; (iii) these functions are orthogonal at least in a Hilbert space $\mathscr H$ that contains $\mathscr A$. This situation turns out to appear in certain time encoders that are part of the increasingly important area of event-based sampling. As a result of pseudo-inversion, we systematically achieve perfect reconstruction whenever the samples uniquely characterize the input, we obtain minimal-norm estimates when the sampling is insufficient, and the reconstruction errors are controlled in the case of noisy sampling. The algorithm consists in alternating two projections according to the general method of projections onto convex sets (POCS) and can be implemented by iterating time-varying discrete-time filtering. We finally show that our signal and sampling assumptions appear in a nontrivial manner in other existing problems of data acquisition. This includes multi-channel time encoding where $\mathscr H$ is of the type $L^2({\mathbb R})^M$, and traditional point sampling with the adoption of a Sobolev space $\mathscr H$.

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arXiv2022

POCS-based framework of signal reconstruction from generalized non-uniform samples

We formalize the use of projections onto convex sets (POCS) for the reconstruction of signals from non-uniform samples in their highest generality. This covers signals in any Hilbert space $\mathscr H$, including multi-dimensional and multi-channel signals, and samples that are most generally inner products of the signals with given kernel functions in $\mathscr H$. An attractive feature of the POCS method is the unconditional convergence of its iterates to an estimate that is consistent with the samples of the input, even when these samples are of very heterogeneous nature on top of their non-uniformity, and/or under insufficient sampling. Moreover, the error of the iterates is systematically monotonically decreasing, and their limit retrieves the input signal whenever the samples are uniquely characteristic of this signal. In the second part of the paper, we focus on the case where the sampling kernel functions are orthogonal in $\mathscr H$, while the input may be confined in a smaller closed space $\mathscr A$ (of bandlimitation for example). This covers the increasingly popular application of time encoding by integration, including multi-channel encoding. We push the analysis of the POCS method in this case by giving a special parallelized version of it, showing its connection with the pseudo-inversion of the linear operator defined by the samples, and giving a multiplierless discrete-time implementation of it that paradoxically accelerates the convergence of the iteration.

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arXiv2022

Bandlimited signal reconstruction from leaky integrate-and-fire encoding using POCS

Leaky integrate-and-fire (LIF) encoding is a model of neuron transfer function in biology that has recently attracted the attention of the signal processing and neuromorphic computing communities as a technique of event-based sampling for data acquisition. While LIF enables the implementation of analog-circuit signal samplers of lower complexity and higher accuracy simultaneously, the core difficulty of this technique is the retrieval of an input from its LIF-encoded output. In this article, we study this problem in the context of bandlimited inputs, by extracting the most abstract features of an LIF encoder as a generalized nonuniform sampler. In this view, the LIF output is seen as the transformation of the input by a known linear operator. We show that the signal reconstruction method of projection onto convex sets (POCS) converges to a weighted pseudo-inverse of this operator. This allows perfect recovery under uniqueness of reconstruction, minimum-norm reconstruction under incomplete sampling, as well as a noise shaping of time quantization that outperforms standard pseudo-inversion. On the practical side, a single iteration of the POCS method can be used to improve any estimate whose LIF samples are not consistent with those of the input, and a rigorous discrete-time implementation of this iteration is proposed that does not require a Nyquist-rate representation of the signals.

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arXiv2020

Time encoding of bandlimited signals: reconstruction by pseudo-inversion and time-varying multiplierless FIR filtering

We propose an entirely redesigned framework of bandlimited signal reconstruction for the time encoding machine (TEM) introduced by Lazar and Tóth. As the encoding part of TEM consists in obtaining integral values of a bandlimited input over known time intervals, it theoretically amounts to applying a known linear operator on the input. We then approach the general question of signal reconstruction by pseudo-inversion of this operator. We perform this task numerically and iteratively using projections onto convex sets (POCS). The algorithm can be implemented exactly in discrete time with multiplications that are all reduced to scaling by signed powers of two, thanks to the use of relaxation coefficients. Meanwhile, the algorithm achieves a rate of convergence similar to that of Lazar and Tóth. For real-time processing, we propose an approximate time-varying FIR implementation, which avoids the splitting of the input into blocks. We finally propose some preliminary analysis of semi-convergence of the algorithm under data noise.

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arXiv2019

Unrestricted iterations of relaxed projections in Hilbert space: Regularity, absolute convergence, and statistics of displacements

Given a finite collection $\mathbf{V}:=(V_1,\dots,V_N)$ of closed linear subspaces of a real Hilbert space $H$, let $P_i$ denote the orthogonal projection operator onto $V_i$ and $P_{i,λ}:= (1-λ)I + λP_i$ denote its relaxation with parameter $λ\in [0,2]$, $i=1,\dots,N$. Under a mild regularity assumption on $\mathbf{V}$ known as `innate regularity' (which, for example, is always satisfied if each $V_i$ has finite dimension or codimension), we show that all trajectories $(x_n)_{0}^\infty$ resulting from the iteration $x_{n+1} := P_{i_n,λ_n}(x_n)$, where the $i_n$ and the $λ_n$ are unrestricted other than the assumption that $\{λ_n : n \in \mathbb{N}\} \subset [η,2{-}η]$ for some $η\in (0,1]$, possess uniformly bounded displacement moments of arbitrarily small orders. In particular, we show that $$ \sum_{n=0}^\infty \|x_{n+1} - x_n \|^γ\leq C \|x_0\|^γ~\mbox{ for all }~ γ> 0,$$ where $C:=C(\mathbf{V},η,γ)<\infty$. This result strengthens prior results on norm convergence of these trajectories, known to hold under the same regularity assumption. For example, with $γ=1$, it follows that the displacements series $\sum (x_{n+1}-x_n)$ converges absolutely in $H$. Quantifying the constant $C(\mathbf{V},η,γ)$, we also derive an effective bound on the distribution function of the norms of the displacements (normalized by the norm of the initial condition) which yields a root-exponential type decay bound on their decreasing rearrangement, again uniformly for all trajectories.

math.FA↗

arXiv2003

Ergodic Dynamics in Sigma-Delta Quantization: Tiling Invariant Sets and Spectral Analysis of Error

This paper has two themes that are intertwined: The first is the dynamics of certain piecewise affine maps on the Euclidean space that arise from a class of analog-to-digital conversion methods called Sigma-Delta quantization. The second is the analysis of reconstruction error associated to each such method. Sigma-Delta quantization generates approximate representations of functions by sequences that lie in a restricted set of discrete values. These are special sequences in that their local averages track the function values closely, thus enabling simple convolutional reconstruction. In this paper, we are concerned with the approximation of constant functions only, a basic case that presents surprisingly complex behavior. An m'th order Sigma-Delta scheme with input x can be translated into a dynamical system that produces a discrete-valued sequence (in particular, a 0-1 sequence) q as its output. When the schemes are stable, we show that the underlying piecewise affine maps possess invariant sets that tile the Euclidean space up to a finite multiplicity. When this multiplicity is one (the single-tile case), the dynamics within the tile is isomorphic to that of a generalized skew translation on the m dimensional torus. The value of x can be approximated using any consecutive M elements in q with increasing accuracy in M. We show that the asymptotical behavior of reconstruction error depends on the regularity of the invariant sets, the order m, and some arithmetic properties of x. We determine the behavior in a number of cases of practical interest and provide good upper bounds in some other cases when exact analysis is not yet available.

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