Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $\alpha$-helices, $\beta$-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.

As machine learning powered decision-making becomes increasingly important in our daily lives, it is imperative to strive for fairness in the underlying data processing. We propose a pre-processing algorithm for fair data representation via which supervised learning results in estimations of the Pareto frontier between prediction error and statistical disparity. In particular, the present work applies the optimal affine transport to approach the post-processing Wasserstein barycenter characterization of the optimal fair $L^2$-objective supervised learning via a pre-processing data deformation. Furthermore, we show that the Wasserstein geodesics from the conditional (on sensitive information) distributions of the learning outcome to their barycenter characterize the Pareto frontier between $L^2$-loss and the average pairwise Wasserstein distance among sensitive groups on the learning outcome. Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen data; (2) the fair representation protects the sensitive information by limiting the inference capability of the remaining data with respect to the sensitive data; (3) the optimal affine maps are computationally efficient even for high-dimensional data.

Privacy-preserving data analysis is emerging as a challenging problem with far-reaching impact. In particular, synthetic data are a promising concept toward solving the aporetic conflict between data privacy and data sharing. Yet, it is known that accurately generating private, synthetic data of certain kinds is NP-hard. We develop a statistical framework for differentially private synthetic data, which enables us to circumvent the computational hardness of the problem. We consider the true data as a random sample drawn from a population$\Omega $according to some unknown density. We then replace$\Omega $by a much smaller random subset$\Omega ^{\ast}$, which we sample according to some known density. We generate synthetic data on the reduced space$\Omega ^{\ast}$by fitting the specified linear statistics obtained from the true data. To ensure privacy we use the common Laplacian mechanism. Employing the concept of Rényi condition number, which measures how well the sampling distribution is correlated with the population distribution, we derive explicit bounds on the privacy and accuracy provided by the proposed method.

Spectral clustering has become one of the most popular algorithms in data clustering and community detection. We study the performance of classical two-step spectral clustering via the graph Laplacian to learn the stochastic block model. Our aim is to answer the following question: when is spectral clustering via the graph Laplacian able to achieve strong consistency, i.e., the exact recovery of the underlying hidden communities? Our work provides an entrywise analysis (an $\ell_{\infty}$-norm perturbation bound) of the Fiedler eigenvector of both the unnormalized and the normalized Laplacian associated with the adjacency matrix sampled from the stochastic block model. We prove that spectral clustering is able to achieve exact recovery of the planted community structure under conditions that match the information-theoretic limits.

Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called self-calibration, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup ${y} = \mathcal{A}({d}) {x} + {\epsilon}$ where only partial information about the sensing matrix $\mathcal{A}({d})$ is known and where $\mathcal{A}({d})$ linearly depends on ${d}$. The goal is to estimate the calibration parameter ${d}$ (resolve the uncertainty in the sensing process) and the signal/object of interest ${x}$ simultaneously. For three different models of practical relevance, we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a spectral method, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques, and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed, and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

Suppose that we have r sensors and each one intends to send a fun9810427ction g (e.g., a signal or an image) to a receiver common to all r sensors. During transmission, each g gets convolved with a function fi. The receiver records the function y, given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals g and the blurring functions f from just one received signal y? This challenging problem, which intertwines blind deconvolution with blind demixing, appears in a variety of applications, such as audio processing, image processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a central role in connection with the future Internet-of-Things. We will prove that under reasonable and practical assumptions, it is possible to solve this, otherwise, highly ill-posed problem and recover the r transmitted functions giand the impulse responses fiin a robust, reliable, and efficient manner, from just one single received function y by solving a semidefinite program. We derive explicit bounds on the number of measurements needed for successful recovery and prove that our method is robust in the presence of noise. Our theory is actually suboptimal, since numerical experiments demonstrate that, quite remarkably, recovery is still possible if the number of measurements is close to the number of degrees of freedom.

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging, and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that a complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.

We consider linear time-varying channels with additive white Gaussian noise. For a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time-varying transfer function and, thus, provide a theoretical justification for a relationship that has been frequently observed in the literature. We then use this eigenvalue estimate to derive an estimate of the mutual information of the channel. Our approach is constructive and is based on a careful balance of the tradeoff between approximate operator diagonalization, signal dimension loss, and accuracy of eigenvalue estimates.

The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The predictions of theorems are confirmed by numerical simulations.

Multiple subcarrier modulation is an appealing scheme for high-data rate optical communication. However a major drawback is its low average power efficiency. While subcarrier reservation is a common approach to combat this problem, little is known about the performance of algorithms that utilize subcarrier reservation. By combining properties of sparse signals with an abstract form of the Uncertainty Principle related to multiple subcarrier signals, we design an effective iterative method for constructing average-power-efficient multicarrier signals. Unlike most existing subcarrier reservation methods, our method provides a guaranteed bound for the achievable average power reduction as well as guaranteed rates of convergence. Numerical simulations demonstrate the performance of the proposed method.

A stylized compressed sensing radar is proposed in which the time- frequency plane is discretized into an N times N grid. Assuming the number of targets K is small (i.e., K Lt N2), then we can transmit a sufficiently "incoherent" pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel compressed sensing approach offers great potential for better resolution over classical radar.

In order to reduce the inter-carrier interference (ICI) produced by frequency offset in OFDM systems, we set up an optimization problem to find the transmission pulse which maximizes the signal-to-average-ICI power ratio. Furthermore, by solving a constrained convex optimization problem, our pulses can satisfy various decay requirements, for instance with respect to a prescribed spectral mask. The latter is an important feature in order to comply with industry standard specifications. Furthermore we discuss issues concerning the numerical treatment of the constrained convex optimization problem. Simulation results show that our pulses outperform many currently known transmission pulses with respect to the ICI criterion.

A bandlimited signal can be reconstructed from its periodic nonuniformly spaced samples provided the average sampling rate is at least the Nyquist rate. Unlike many previously published methods, the algorithm derived in this paper is designed that pays special attention to various practical constraints. In particular, we propose a fast and numerically robust reconstruction method which can utilize FIR filters with a small number of taps and requires only a modest amount of oversampling to achieve high accuracy. The efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure combined with utilizing localized Fourier analysis. We discuss applications in time-interleaved analog-to-digital converters where nonuniform periodic sampling arises due to timing mismatches. Finally, numerical simulations demonstrate the performance of our algorithm.

We present a digital background technique for correcting the time and gain mismatches in a time-interleaved analog-to-digital converter (ADC) system. The proposed blind calibration is applicable to any number of time-interleaved ADCs and requires only modest oversampling. Simulation results show fast convergence and desirable detection accuracy. After the mismatch errors detection, the resulting signal to noise ratio (SNR) of the output signal is shown to be higher than the SNR of the input signal in a 16-ADC system.

Sum capacity optimal signatures in synchronous code-division multiple-access (CDMA) systems are functions of the codebook length as well as the number of active users. A new signature set must be assigned every time the number of active users changes. This correspondence considers signature sets that are less sensitive to changes in the number of active users. Equiangular signature sequences are proven to solve a certain max-min signal-to-interference-plus-noise problem, which results from their interference invariance. Unions of orthonormal bases have subsets that come close to satisfying the Welch bound. Bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided. Connections are made between these signature design problems, Grassmannian line packing, frame theory, and algebraic geometry

Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm.

We propose to apply a technique called time-reversal to UWB communications. In time-reversal a signal is precoded such that it focuses both in time and in space at a particular receiver. Spatial focusing reduces interference to other co-existing systems. Due to temporal focusing, the received power is concentrated within a few taps and the task of equalizer design becomes much simpler than without focusing. Furthermore, temporal focusing allows a large increase in transmission rate compared to schemes that let the impulse response ring out before the next symbol is sent. Our paper introduces time-reversal, investigates the benefit of temporal focusing, and examines the performance of an MMSE-TR equalizer in an UWB channel.

Codebooks constructed from Welch bound equality (WBE) sequences have been show to be optimal in terms of sum capacity in synchronous CDMA systems. Unfortunately, these codebooks are a function of the number of active signatures and need to be reassigned as the number of active users changes to maintain optimality. To mitigate the problems caused by the loss of the Welch bound equality property, in this paper we propose a special subclass of WBE sequences for which the interference power experienced by each user depends only on the number of active users and the dimensions of the code. In deference to the relationship with Grassmannian line packing, we refer to this as a Grassmannian signature set. We study the interference properties of this set, comment on the sequence design problem, and illustrate improvements over arbitrary WBE sequence sets via simulation.

Multiple-input multiple-output (MIMO) wireless systems provides capacity much larger than that provided by traditional single-input single-output (SISO) wireless systems. Beamforming is a low complexity technique that increases the receive signal-to-noise ratio (SNR), however, it requires channel knowledge. Since in practice channel knowledge at the transmitter is difficult to realize, we propose a technique where the receiver designs the beamforming vector and sends it to the transmitter by transmitting a label in a finite set, or codebook, of beamforming vectors. A codebook design method for quantized versions of maximum ratio transmission, equal gain transmission, and generalized selection diversity with maximum ratio combining at the receiver is presented. The codebook design criterion exploits the quantization problem's relationship with Grassmannian line packing. Systems using the beamforming codebooks are shown to have a diversity order of the product of the number of transmit and the number of receive antennas. Monte Carlo simulations compare the performance of systems using this new codebook method with the performance of systems using previously proposed quantized and unquantized systems.

Transmission over wireless channels is subject to time dispersion due to multipath propagation and to frequency dispersion due to the Doppler effect. Standard orthogonal frequency-division multiplexing (OFDM) systems, using a guard-time interval or cyclic prefix, combat intersymbol interference (ISI), but provide no protection against interchannel interference (ICI). This drawback has led to the introduction of pulse-shaping OFDM systems. We first present a general framework for pulse shape design. Our analysis shows that certain pulse shapes proposed in the literature are, in fact, optimal in a well-defined sense. Furthermore, our approach provides a simple way to adapt the pulse shape to varying channel conditions. We then show that (pulse-shaping) OFDM systems based on rectangular time-frequency lattices are not optimal for time- and frequency-dispersive wireless channels. This motivates the introduction of lattice-OFDM (LOFDM) systems which are based on general time-frequency lattices. Using results from sphere packing theory, we show how to design LOFDM systems (lattice and pulse shape) optimally for timeand frequency-dispersive channels in order to minimize the joint ISI/ICI. Our theoretical analysis is confirmed by numerical simulations, showing that LOFDM systems outperform traditional pulse-shaping OFDM systems with respect to robustness against ISI/ICI.

Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple-input multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector; both are hard to realize. In this article, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent and identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.

Standard OFDM systems are associated with a rectangular grid in the time-frequency plane. However such a setup is in general not optimal for pulse shaping OFDM systems for doubly dispersive channels. We introduce lattice-OFDM systems (LOFDM), which are OFDM systems constructed with respect to general lattices in the time-frequency plane. We show how to design optimal pulse shapes for LOFDM systems. Furthermore we demonstrate by theoretical considerations and numerical simulations that LOFDM systems using hexagonal-type lattices outperform ordinary OFDM systems with regard to robustness against ISI/ICI.

In many applications signals can only be sampled at nonuniformly spaced points. For a reliable reconstruction of the signal from its samples we require knowledge of the bandwidth of the signal, which however is often not known a priori. Therefore robust and efficient methods are needed that allow one to estimate the bandwidth of a signal from nonuniform spaced, noisy samples. We present two procedures for bandwidth estimation. The first method is based on the discrete Bernstein inequality and Newton's divided differences and is computationally very efficient. The second method requires somewhat more computational effort, since it simultaneously estimates the bandwidth and provides a reconstruction of the signal. It is based on a multi-scale conjugate gradient algorithm for the solution of a nested sequence of Toeplitz systems and is particularly useful in case of noisy data. Examples from various applications demonstrate the performance of the proposed methods.

An efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed. The novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two-dimensional (2-D) trigonometric polynomials, where the matrix is of block-Toeplitz type with Toeplitz blocks. This system is solved iteratively by the conjugate gradient (CG) method. We show that the use of so-called adaptive weights in the establishment of the block Toeplitz matrix can be seen as efficient preconditioning. The superiority of the new method over conventional approaches is demonstrated by numerical experiments.

Diagnostics in medicine is often based on analysis of medical images. Extraction of the shape of objects is one basic step for further image analysis. It consists of two steps: (i) edge detection and (ii) contour recovery based on the detected edge points. Here, the authors focus on the second step. They describe the boundary of an object by trigonometric polynomials and present a fast and robust method for the approximation of the boundary from a given set of nonuniformly distributed noisy edge points. The authors show how the proposed technique can be efficiently generalized to 3-D surface recovery. Examples from 2-D and 3-D echocardiography are given.