arXiv subjects

Peter Balazs

Publications and source records attributed to Peter Balazs.

At least 37 records · Page 2Linked to original sources

arXiv2023

Time-frequency analysis on flat tori and Gabor frames in finite dimensions

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori \begin{equation*} \mathbb{T}_{N}^2=\mathbb{R}^2/(\mathbb{Z}\times N\mathbb{Z})=[0,1]\times \lbrack 0,N] \end{equation*} act as phase spaces. We work on an $N$-dimensional subspace $S_{N}$ of distributions periodic in time and frequency in the dual $S_0'(\mathbb{R})$ of the Feichtinger algebra $S_0(\mathbb{R})$ and equip it with an inner product. To construct the Hilbert space $S_{N}$ we apply a suitable double periodization operator to $S_0(\mathbb{R})$. On $S_{N}$, the STFT is applied as the usual STFT defined on $S_0'(\mathbb{R})$. This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallst{é}n for Gabor frames with Gaussian windows and which, for $N$ odd, produces an explicit \emph{full spark Gabor frame}. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

math.FA↗

arXiv2023

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are naturally linked to bounded operators. To tackle unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of those sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, like e.g. a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

math.FA↗

arXiv2023

Feasibility of the EDICAM camera for runaway electron detection in JT-60SA disruptions

The visible camera system EDICAM (Event Detection Intelligent Camera), recently installed on JT-60SA, is simulated to assess whether it can be used for measuring synchrotron radiation from relativistic runaway electrons. In this simulation, the SOFT synthetic synchrotron diagnostic framework is used to compute the synthetic synchrotron camera images from a JT-60SA-like disruption modelled with the DREAM disruption simulation code. In the studied scenario, a large amount of argon is added to the plasma, and a disruption is simulated by starting a prescribed exponential temperature drop and finishing with further cooling provided by the argon in a self-consistent simulation of the current quench. The background plasma evolution is calculated by DREAM self-consistently with the fast electron population, which is modelled kinetically. The resulting runaway electron distribution function along with the parameters of the EDICAM visible camera system are used as an input to the SOFT framework to assess the feasibility of the camera for runaway electron detection. We find that the runaway electron beam formed in the disruption can produce synchrotron radiation observable by the EDICAM system, thus enabling the use of the EDICAM for the characterization of runaway electron beams.

physics.plasm-ph↗

arXiv2023

Fitting Auditory Filterbanks with Multiresolution Neural Networks

Waveform-based deep learning faces a dilemma between nonparametric and parametric approaches. On one hand, convolutional neural networks (convnets) may approximate any linear time-invariant system; yet, in practice, their frequency responses become more irregular as their receptive fields grow. On the other hand, a parametric model such as LEAF is guaranteed to yield Gabor filters, hence an optimal time-frequency localization; yet, this strong inductive bias comes at the detriment of representational capacity. In this paper, we aim to overcome this dilemma by introducing a neural audio model, named multiresolution neural network (MuReNN). The key idea behind MuReNN is to train separate convolutional operators over the octave subbands of a discrete wavelet transform (DWT). Since the scale of DWT atoms grows exponentially between octaves, the receptive fields of the subsequent learnable convolutions in MuReNN are dilated accordingly. For a given real-world dataset, we fit the magnitude response of MuReNN to that of a well-established auditory filterbank: Gammatone for speech, CQT for music, and third-octave for urban sounds, respectively. This is a form of knowledge distillation (KD), in which the filterbank ''teacher'' is engineered by domain knowledge while the neural network ''student'' is optimized from data. We compare MuReNN to the state of the art in terms of goodness of fit after KD on a hold-out set and in terms of Heisenberg time-frequency localization. Compared to convnets and Gabor convolutions, we find that MuReNN reaches state-of-the-art performance on all three optimization problems.

cs.SD↗

arXiv2023

Convex Geometry of ReLU-layers, Injectivity on the Ball and Local Reconstruction

The paper uses a frame-theoretic setting to study the injectivity of a ReLU-layer on the closed ball of $\mathbb{R}^n$ and its non-negative part. In particular, the interplay between the radius of the ball and the bias vector is emphasized. Together with a perspective from convex geometry, this leads to a computationally feasible method of verifying the injectivity of a ReLU-layer under reasonable restrictions in terms of an upper bound of the bias vector. Explicit reconstruction formulas are provided, inspired by the duality concept from frame theory. All this gives rise to the possibility of quantifying the invertibility of a ReLU-layer and a concrete reconstruction algorithm for any input vector on the ball.

cs.LG↗

arXiv2023

Accelerated Griffin-Lim algorithm: A fast and provably converging numerical method for phase retrieval

The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. In this paper, we present a fast inertial/momentum based algorithm for the phase retrieval problem and we prove a convergence guarantee for the new algorithm and for the Fast Griffin-Lim algorithm, whose convergence remained unproven in the past decade. In the final chapter, we compare the algorithm for the Short Time Fourier transform phase retrieval with the Griffin-Lim algorithm and FGLA and to other iterative algorithms typically used for this type of problem.

math.OC↗

arXiv2023

Faster than Fast: Accelerating the Griffin-Lim Algorithm

The phase retrieval problem is found in various areas of applications of engineering and applied physics. It is also a very active field of research in mathematics, signal processing and machine learning. In this paper, we present an accelerated version of the well known Fast Griffin-Lim algorithm (FGLA) for the phase retrieval problem in a general setting. It has increased the speed of convergence, and most importantly, the limit points of the generated sequence can reach a significantly smaller error than the ones generated by FGLA. We will give a motivation of the acceleration and compare it numerically to its predecessors and other algorithms typically used to solve similar problems.

math.OC↗

arXiv2023

On the relation of the frame-related operators of fusion frame systems

Frames have been investigated frequently over the last few decades due to their valuable properties, which are desirable for various applications as well as interesting for theory. Some applications additionally require distributed processing techniques, which naturally leads to the concept of fusion frames and fusion frame systems. The latter consists of a system of subspaces, equipped with local frames on each of them, and a global frame. In this paper, we investigate the relations of the associated frame-related operators on all those three levels. For that we provide a detailed investigation on bounded block diagonal operators between Hilbert direct sums. We give the relation of the frame-related operators of the fusion frame and the corresponding frame systems in terms of operator identities. By applying these identities we prove further properties of fusion frame systems.

math.FA↗

arXiv2023

A Survey of Fusion Frames in Hilbert Spaces

Fusion frames are a very active area of research today because of their myriad of applications in pure mathematics, applied mathematics, engineering, medicine, signal and image processing and much more. They provide a great flexibility for designing sets of vectors for applications and are therefore prominent in all these areas, including e.g. mitigating the effects of noise in a signal or giving robustness to erasures. In this chapter, we present the fundamentals of fusion frame theory with an emphasis on their delicate relation to frame theory. The goal here is to provide researchers and students with an easy entry into this topic. Proofs for fusion frames will be self-contained and differences between frames and fusion frames are analyzed. In particular, we focus on the subtleties of fusion frame duality. We also provide a reproducible research implementation.

math.FA↗

arXiv2022

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant $κ>0$ so that for all $C,β>0$ and $N\in\mathbb{N}$ the following is true. Let $(x_j)_{j=1}^N$ and $(f_j)_{j=1}^N$ be sequences in a finite dimensional Hilbert space which satisfy $\|x_j\|=\|f_j\|$ for all $1\leq j\leq N$ and $$\Big\|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big\|\leq C\|x\|, \qquad\textrm{ for all $x\in \ell_2^M$ and $|\varepsilon_j|=1$}. $$ If the frame operator for $(f_j)_{j=1}^N$ has eigenvalues $λ_1\geq...\geqλ_M$ and $λ_1\leq βM^{-1}\sum_{j=1}^Mλ_j$ then $(f_j)_{j=1}^N$ has Bessel bound $κβ^2 C$. The same holds for $(x_j)_{j=1}^N$.

math.FA↗

arXiv2022

The affine ensemble: determinantal point processes associated with the ax+b group

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane C^+ associated with the ax+b (affine) group, depending on an admissible Hardy function ψ. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set Ω contained in C^+. As a special case one recovers the DPP related to the weighted Bergman kernel. When ψ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.

math.PR↗

arXiv2022

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

We study the connection between STFT multipliers $A^{g_1,g_2}_{1\otimes m}$ having windows $g_1,g_2$, symbols $a(x,ω)=(1\otimes m)(x,ω)=m(ω)$, $(x,ω)\in\mathbb{R}^{2d}$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\mathbb{R}^d$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $g_1,g_2$ for the equality $T_{m_2}= A^{g_1,g_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{g_1,g_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A^{g_1,g_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{g,g}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

math.FA↗

arXiv2022

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties, as completeness or the property of being a (semi-)frame, of a certain sequence in one of the spaces propagate to other spaces in a scale. We link that to the properties of the respective frame-related operators, like analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered, in particular that the upper semi-frame property is kept in larger spaces, while the lower one to smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces are not equal.

math.FA↗

arXiv2022

Fast Matching Pursuit with Multi-Gabor Dictionaries

Finding the best K-sparse approximation of a signal in a redundant dictionary is an NP-hard problem. Suboptimal greedy matching pursuit (MP) algorithms are generally used for this task. In this work, we present an acceleration technique and an implementation of the matching pursuit algorithm acting on a multi-Gabor dictionary, i.e., a concatenation of several Gabor-type time-frequency dictionaries, each of which consisting of translations and modulations of a possibly different window and time and frequency shift parameters. The technique is based on pre-computing and thresholding inner products between atoms and on updating the residual directly in the coefficient domain, i.e., without the round-trip to the signal domain. Since the proposed acceleration technique involves an approximate update step, we provide theoretical and experimental results illustrating the convergence of the resulting algorithm. The implementation is written in C (compatible with C99 and C++11) and we also provide Matlab and GNU Octave interfaces. For some settings, the implementation is up to 70 times faster than the standard Matching Pursuit Toolkit (MPTK).

math.NA↗

arXiv2022

Audio Inpainting via $\ell_1$-Minimization and Dictionary Learning

Audio inpainting refers to signal processing techniques that aim at restoring missing or corrupted consecutive samples in audio signals. Prior works have shown that $\ell_1$- minimization with appropriate weighting is capable of solving audio inpainting problems, both for the analysis and the synthesis models. These models assume that audio signals are sparse with respect to some redundant dictionary and exploit that sparsity for inpainting purposes. Remaining within the sparsity framework, we utilize dictionary learning to further increase the sparsity and combine it with weighted $\ell_1$-minimization adapted for audio inpainting to compensate for the loss of energy within the gap after restoration. Our experiments demonstrate that our approach is superior in terms of signal-to-distortion ratio (SDR) and objective difference grade (ODG) compared with its original counterpart.

cs.SD↗

arXiv2022

Phase-Based Signal Representations for Scattering

The scattering transform is a non-linear signal representation method based on cascaded wavelet transform magnitudes. In this paper we introduce phase scattering, a novel approach where we use phase derivatives in a scattering procedure. We first revisit phase-related concepts for representing time-frequency information of audio signals, in particular, the partial derivatives of the phase in the time-frequency domain. By putting analytical and numerical results in a new light, we set the basis to extend the phase-based representations to higher orders by means of a scattering transform, which leads to well localized signal representations of large-scale structures. All the ideas are introduced in a general way and then applied using the STFT.

cs.SD↗

arXiv2022

Continuous frames in tensor product Hilbert spaces, localization operators and density operators

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces.

math.FA↗

arXiv2020

Representation of Operators Using Fusion Frames

For finding the numerical solution of operator equations in many applications a decomposition in subspaces is needed. Therefore, it is necessary to extend the known method of matrix representation to the utilization of fusion frames. In this paper we investigate this representation of operators on a Hilbert space $\Hil$ with Bessel fusion sequences, fusion frame and Riesz decompositions. We will give the basic definitions. We will show some structural results and give some examples. Furthermore, in the case of Riesz decompositions, we prove that those functions are isomorphisms. Also, we want to find the pseudo-inverse and the inverse (if there exists) of such matrix representations. We are going to apply this idea to the Schatten $p$-class operators. Finally, we show that tensors of fusion frame are frames in the space of Hilbert-Schmidt operators.

math.FA↗