One-Bit Tensor Sensing: Algorithms & Guarantees

Updated 22 October 2025

One-bit tensor sensing is the recovery of structured high-dimensional tensors using sign-only measurements that capture directional information.
It employs tailored measurement schemes such as random dithering and structural constraints (sparsity, low-rank) to mitigate severe quantization losses.
Advanced recovery algorithms, including convex relaxation and deep unrolling, provide provable guarantees and scalable performance in practical applications.

One-bit tensor sensing refers to the recovery of structured high-dimensional arrays (tensors) from extremely coarse, sign-only (1-bit) measurements. This paradigm arises at the intersection of compressive sensing, high-order signal recovery, and quantized data acquisition, and leverages both algorithmic and geometric tools to address the challenges introduced by tensor structure and severe quantization.

1. Fundamental Concepts and Mathematical Principles

In one-bit tensor sensing, each linear or multilinear measurement of a tensor $\mathcal{T}$ is quantized to a single bit, typically retaining only the sign: $y_i = \operatorname{sign}(\langle \mathcal{A}_i, \mathcal{T} \rangle)$ where $\mathcal{A}_i$ denotes a possibly random sensing tensor and $\langle \cdot, \cdot \rangle$ is the natural inner product over the tensor domain. This extends the vector one-bit measurement model $y = \operatorname{sign}(A x)$ into higher-dimensional settings, allowing the measurement process to capture multi-modal correlations.

Key properties and challenges arise because the sign operator is both highly nonlinear and information-losing: only directional information is retained, and amplitude is completely discarded. For tensor problems, additional structure—such as sparsity along modes or low-rankness—enables recovery from sign-only observations, provided suitable measurement operators are used and structural priors enforced (Ghadermarzy et al., 2018).

Central theoretical concepts include:

Restricted Isometry Property (RIP) for one-bit embeddings: For class $\mathcal{K}$ of structured tensors, the sign map should preserve angular/metric geometry up to distortion; for vectors, this is

$\sup_{x, x' \in \mathcal{K}} |\operatorname{d}_H(\operatorname{sign}(Ax), \operatorname{sign}(Ax')) - d(x, x')| < \delta,$

where $d_H$ is Hamming distance and $d(\cdot,\cdot)$ is an appropriate geometric (e.g., normalized geodesic) distance (Bilyk et al., 2015, Bilyk et al., 2015).

Order-Optimal Sample Complexity: The number of one-bit measurements required for robust recovery of an underlying order- $d$ tensor is often $O(r^{d^2-d}N)$ for rank- $r$ tensors in $\mathbb{R}^{N\times N\times\cdots\times N}$ , which matches unquantized settings up to constants when $r=O(1)$ (Ghadermarzy et al., 2018).

2. Signal Models and Measurement Schemes

One-bit tensor sensing applies to tensors with sparsity, low-rank, or joint structural constraints. Canonical models include:

Sparse Tensors: Nonzero entries concentrated in a small $\ell_0$ set across one or more modes; support recovery and amplitude estimation are of primary interest (Gupta et al., 2015, Acharya et al., 2017).
Low-Rank Tensors: Generalization of matrix rank; e.g., via CP or Tucker models; regularized by quasi-norms (max-qnorm) or convex surrogates (atomic M-norm) (Ghadermarzy et al., 2018).
Joint Sparse Structures: Multiple measurement vectors or sensor observations constrained to share the same support structure, analyzed in multi-dimensional settings (Gupta et al., 2015).

Measurement operators can be random (i.i.d. Gaussian, $\alpha$ -stable, or time-varying dithered (Eamaz et al., 2023)), deterministic combinatorial, or tailored to be universal across all signals in the model class.

For quantization, random dithering—addition of a randomly chosen threshold before sign quantization—is often employed to enable unbiased estimation and improve performance in both vector and tensor cases (Eamaz et al., 2023, Yeganegi et al., 15 May 2024).

3. Recovery Algorithms and Optimization Techniques

Recovery from one-bit measurements necessitates algorithms capable of handling both the nonlinear data model and tensor structure:

Convex Relaxation: For sparse or low-rank tensors, minimize an objective such as the $\ell_{1}$ norm (for sparsity) or nuclear/max-qnorm (for low-rankness) subject to sign constraints,

$\min_X\quad \text{Regularizer}(X) \quad \text{such that} \quad \operatorname{sign}(\mathcal{A}(X)) = y \text{ (or } y_i \langle \mathcal{A}_i, X \rangle \ge 0)$

or, with likelihood modeling, negative log-likelihood functions are used (Gupta et al., 2015, Ghadermarzy et al., 2018).

Iterative Shrinkage and Thresholding (ISTA) and Deep Unrolling: Algorithms like ISTA are unrolled into deep networks (e.g., LISTA) for efficient and learnable one-bit sparse/tensor estimation, with layer-dependent learned parameters yielding improved performance (Yeganegi et al., 15 May 2024, Khobahi et al., 2019).
Nuclear Norm and Singular Value Thresholding (SVT): For low-rank (particularly matrix) completion from one-bit data, SVT and its variants (OB-SVT) solve convex relaxations with linear inequality constraints derived from quantized samples (Eamaz et al., 2023).
One-Scan and Blockwise Algorithms: Exploiting $\alpha$ -stable projections and blockwise structure allows "one-pass" decoding and efficient computation (Li, 2015, Eamaz et al., 2023).
Kaczmarz-Type Methods with Feasibility Polyhedra: With sample abundance, feasibility is posed as a large system of linear inequalities, with randomized or preconditioned Kaczmarz methods offering scalable reconstruction (Eamaz et al., 2023).

4. Theoretical Guarantees and Geometric Insights

The effectiveness of one-bit tensor sensing is supported by a combination of metric embedding theory, combinatorial design, and statistical learning bounds:

Restricted Isometry: For $s$ -sparse vectors or their tensor generalizations, $m \sim \delta^{-2} s \log(n/s)$ one-bit measurements suffice for $\delta$ -RIP (matching linear unquantized settings) (Bilyk et al., 2015).
Combinatorial Structures: For universal recovery, measurement schemes explicitly constructed via combinatorial objects such as union free families provide near-optimal measurement bounds; for example, $O(k^2 \log n)$ or $O(k^{3/2} \log n)$ for support recovery in the $k$ -sparse case (Acharya et al., 2017, Mazumdar et al., 2021, Bansal et al., 2022).
Geometric Discrepancy and Stolarsky Principle: Embedding the sphere into the Hamming cube via sign-linear maps induces a wedge discrepancy, with dimension-corrected rates; the Stolarsky invariance principle links $L^2$ average embedding error to point-set energies, guiding optimal hyperplane placement and matrix design (Bilyk et al., 2015).
Sample Complexity in Tensors: The number of binary measurements for rank- $r$ , order- $d$ tensors in $\mathbb{R}^{N \times\cdots\times N}$ is $O(Nd)$ for $r=O(1)$ , provided appropriate max-qnorm or M-norm regularization (Ghadermarzy et al., 2018).

5. Applications and Practical Considerations

One-bit tensor sensing has immediate application in scenarios where hardware or transmission constraints enforce severe quantization:

Context-Aware Recommender Systems: Multimodal tensors encoding user, item, and context can be estimated from 1-bit feedback; tensor methods outperform matricized approaches for rating prediction and binary classification tasks (Ghadermarzy et al., 2018).
Sensor Networks: Distributed sensors exchange only sign-based measurements to a central fusion node; joint-sparse tensor recovery methods allow accurate field estimation with extremely low per-sensor bit rates (Gupta et al., 2015).
Massive MIMO and Radar: For DoA estimation with one-bit ADCs, covariance recovery from dithered 1-bit samples linked to sparse direction estimation is addressed using deep unrolled networks, enabling robust direction finding under minimal hardware (Yeganegi et al., 15 May 2024).
Low-Rank Data Completion: One-bit SVT and maximum likelihood methods are used for one-bit quantized matrix/tensor completion in imaging, recommender, and sensor applications (Eamaz et al., 2023).

Furthermore, the paradigms of one-scan, causal inference, and time-varying dithering (for improved resolution and unbiasedness) have direct hardware implications. In embedded and high-speed systems, quantization of both measurement and reconstruction operators provides substantial resource savings (Feuillen et al., 2020).

6. Advanced Topics: Dithering, Universality, and Extension to Tensors

Sample abundance, or extremely high sampling rates with one-bit ADCs, enables casting nonlinear recovery as feasibility over high-dimensional polyhedra, with convergence governed by properties such as the average distance to randomly dithered hyperplanes (Eamaz et al., 2023). Dithering before quantization is pivotal for unbiased estimation; time-varying and randomized thresholds improve numerical stability and reduce required measurements (Eamaz et al., 2023).

Universal measurement operators—those that enable recovery for all signals in a model class—are analyzed through combinatorial and geometric techniques. Results extend from sparse vectors to low-rank or structured tensors, where tensorization of measurement processes and regularization enables generalization (Ghadermarzy et al., 2018, Bansal et al., 2022).

A plausible implication is that frameworks combining sample abundance, tailored regularization (e.g., atomic M-norm), geometric discrepancy control, and deep unrolled algorithms will underpin further practical advances in one-bit tensor sensing.

7. Limitations and Future Directions

Current limitations include the need to further optimize the measurement complexity (especially reducing logarithmic overheads for high-dimensional tensors), develop robust noisy one-bit tensor recovery in more adversarial conditions, and create practical algorithms for nonsmooth or highly structured tensor spaces. Extending energy-based geometric discrepancy minimization from the sphere to tensor product manifolds may provide new metric embedding insights for future designs (Bilyk et al., 2015).

Emerging directions involve automatic learning of priors and adaptive weighting via deep networks, unsupervised support estimation in tensor settings, and extending feasibility methods to more general classes (non-sparse, multimodal, or correlated noise signals). The hardware-driven constraints (binarization of measurement operators, power-efficient ADCs) continue to motivate both algorithmic and theoretical advances.

In summary, one-bit tensor sensing constitutes a mathematically rich and practically vital area at the interface of high-dimensional signal recovery, geometric embedding theory, combinatorial design, and computational optimization. The field has made significant progress in developing structural recovery guarantees, efficient algorithms, and real-world applications, while multiple avenues for theoretical refinement and practical deployment remain open.