Time for dithering: fast and quantized random embeddings via the restricted isometry property

Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect the restricted isometry property (RIP) can induce a quantized random embedding with controllable multiplicative and additive distortions with respect to the pairwise distances of the data points beings considered. In other words, linear matrices having fast matrix-vector multiplication algorithms (e.g., based on partial Fourier ensembles or on the adjacency matrix of unbalanced expanders) can be readily used in the definition of fast quantized embeddings with small distortions. This implication is made possible by applying right after the linear map an additive and random "dither" that stabilizes the impact of the uniform scalar quantization operator applied afterwards. For different categories of RIP matrices, i.e., for different linear embeddings of a metric space $(\mathcal K \subset \mathbb R^n, \ell_q)$ in $(\mathbb R^m, \ell_p)$ with $p,q \geq 1$, we derive upper bounds on the additive distortion induced by quantization, showing that it decays either when the embedding dimension $m$ increases or when the distance of a pair of embedded vectors in $\mathcal K$ decreases. Finally, we develop a novel "bi-dithered" quantization scheme, which allows for a reduced distortion that decreases when the embedding dimension grows and independently of the considered pair of vectors.Comment: Keywords: random projections, non-linear embeddings, quantization, dither, restricted isometry property, dimensionality reduction, compressive sensing, low-complexity signal models, fast and structured sensing matrices, quantized rank-one projections (31 pages

Quantized Radio Map Estimation Using Tensor and Deep Generative Models

Spectrum cartography (SC), also known as radio map estimation (RME), aims at crafting multi-domain (e.g., frequency and space) radio power propagation maps from limited sensor measurements. While early methods often lacked theoretical support, recent works have demonstrated that radio maps can be provably recovered using low-dimensional models -- such as the block-term tensor decomposition (BTD) model and certain deep generative models (DGMs) -- of the high-dimensional multi-domain radio signals. However, these existing provable SC approaches assume that sensors send real-valued (full-resolution) measurements to the fusion center, which is unrealistic. This work puts forth a quantized SC framework that generalizes the BTD and DGM-based SC to scenarios where heavily quantized sensor measurements are used. A maximum likelihood estimation (MLE)-based SC framework under a Gaussian quantizer is proposed. Recoverability of the radio map using the MLE criterion are characterized under realistic conditions, e.g., imperfect radio map modeling and noisy measurements. Simulations and real-data experiments are used to showcase the effectiveness of the proposed approach.Comment: 16 pages, 9 figure

Location-free Spectrum Cartography

Spectrum cartography constructs maps of metrics such as channel gain or received signal power across a geographic area of interest using spatially distributed sensor measurements. Applications of these maps include network planning, interference coordination, power control, localization, and cognitive radios to name a few. Since existing spectrum cartography techniques require accurate estimates of the sensor locations, their performance is drastically impaired by multipath affecting the positioning pilot signals, as occurs in indoor or dense urban scenarios. To overcome such a limitation, this paper introduces a novel paradigm for spectrum cartography, where estimation of spectral maps relies on features of these positioning signals rather than on location estimates. Specific learning algorithms are built upon this approach and offer a markedly improved estimation performance than existing approaches relying on localization, as demonstrated by simulation studies in indoor scenarios.Comment: 14 pages, 12 figures, 1 table. Submitted to IEEE Transactions on Signal Processin

Radio Map Estimation: A Data-Driven Approach to Spectrum Cartography

Radio maps characterize quantities of interest in radio communication environments, such as the received signal strength and channel attenuation, at every point of a geographical region. Radio map estimation typically entails interpolative inference based on spatially distributed measurements. In this tutorial article, after presenting some representative applications of radio maps, the most prominent radio map estimation methods are discussed. Starting from simple regression, the exposition gradually delves into more sophisticated algorithms, eventually touching upon state-of-the-art techniques. To gain insight into this versatile toolkit, illustrative toy examples will also be presented