Sign-Based Estimation Methods

• Sign-based estimation is a class of methods that uses only sign information to robustly recover parameters in quantized sensing and binary regression models.
• It employs maximum likelihood estimation and convex reformulations to tackle nonlinearity and noise, ensuring consistency and asymptotic efficiency.
• These techniques are crucial in applications like wireless communications and compressed sensing, where measurement constraints require robustness to both additive and multiplicative noise.

Sign-based estimation refers to a wide class of parameter estimation and inferential procedures that utilize sign information—typically, the sign of observations, residuals, or projected data—in place of, or in addition to, their magnitudes. Across statistical signal processing, robust statistics, machine learning, and related areas, sign-based methodologies offer distinct advantages in robustness, computational simplicity, and performance under various noise and structural conditions. Contemporary research encompasses both classical one-bit measurement models and modern extensions introducing perturbations, robustification, and convex reformulations.

1. Fundamental Principles of Sign-Based Estimation

Sign-based estimation typically arises in models where the measurement process yields only binary sign information, as in quantized sensing or 1-bit regression. The canonical measurement model is

$$y = \operatorname{sign}\big((H + E)^\top w + n\big)$$

where $H$ is a known deterministic sensing matrix, $E$ is a random perturbation matrix (with i.i.d. Gaussian entries $e_{ij} \sim \mathcal{N}(0, \sigma_e^2)$), $w$ is the deterministic parameter vector to be estimated, and $n \sim \mathcal{N}(0, \sigma_n^2 I)$ is additive noise.

Sign-based estimators focus on recovering $w$ given only the sign vector $y$. The nonlinearity (from the sign function) and the information loss (magnitude discarded) pose challenges for classical estimation but confer remarkable robustness, especially in heavy-tailed or non-Gaussian noise settings. Furthermore, sign-based methods remain viable under sensing matrix uncertainty, as characterized by the perturbation $E$.

Key features:

• Only the sign of the measurement is used—making the method robust to amplitude outliers.
• The variance of the effective noise becomes

$\sigma_z^2 = \sigma_e^2 \|w\|_2^2 + \sigma_n^2$

so both the perturbation and additive noise contribute to the overall uncertainty.

• In the absence of additive noise ($\sigma_n^2 = 0$), the sign measurements lose information about the scale of $w$; only its direction can be estimated.

2. Maximum Likelihood Estimation and Its Properties

The estimation task is most naturally addressed using maximum likelihood estimation (MLE). The log-likelihood for $N$ sign observations is given by

$$\mathcal{L}(w) = \sum_{i=1}^N \log \Phi\left( \frac{y_i (h_i^\top w)}{\sigma_z} \right)$$

where $\Phi(\cdot)$ is the standard normal cumulative distribution function.

Properties and Theoretical Guarantees:

• Consistency: Under mild conditions (parameter boundedness, continuous distribution for $h_i$), the ML estimator is consistent as $N \to \infty$.
• Identifiability: Requires that $H$ has full row rank for unique recovery.
• Efficiency: Asymptotically, the ML estimator attains the Cramér–Rao lower bound (CRLB).

CRLB: The Fisher information matrix is derived as

$J(w) = M \Lambda M^\top$

where $M = (I - (\sigma_e^2/\sigma_z^2) w w^\top)H$ and $\Lambda$ is a diagonal matrix with entries involving the Gaussian likelihood: $$\lambda_{ii} = \frac{1}{2\pi \sigma_z^2} \left[\frac{1}{1/\Phi(A_i) + 1/\Phi(-A_i)}\right] e^{-A_i^2}$$ with $A_i = (h_i^\top w)/\sigma_z$. The minimum mean-square error (MSE) for unbiased estimation is lower-bounded via $\operatorname{tr}\{(J(w))^{-1}\}$.

3. Impact of Sensing Matrix Perturbation

The presence of Gaussian perturbation in the sensing matrix fundamentally changes both estimation accuracy and statistical properties:

• Perturbation Degrades Estimation: As the perturbation strength $\sigma_e^2$ grows, the effective noise increases, and estimator performance, measured in MSE, worsens. The CRLB shows linear or quadratic scaling in the relative noise ratio $\gamma = \sigma_e^2 \|w\|_2^2/\sigma_n^2$.
• Perturbation Can Occasionally Help: In low-noise scenarios, a moderate randomization from $E$ may enhance informative randomness and thus marginally improve estimation under specific regimes.
• Scale Indeterminacy Without Additive Noise: If only multiplicative perturbation is present ($\sigma_n^2 = 0$), the estimator can only recover the direction of $w$ due to the invariance of the sign under scaling.
• Bias in Magnitude: Neglecting matrix perturbation ($\sigma_e^2 = 0$) biases the magnitude of the parameter estimate, although the direction remains correct:

$$w_{\text{ignored}} = \frac{w_{\text{ML}}}{\sqrt{1 + (\sigma_e^2/\sigma_n^2) \|w_{\text{ML}}\|_2^2}}$$

indicating that sign-based inference can still provide meaningful direction estimates even when perturbation is not modeled explicitly.

4. Convex Reformulation and Computational Aspects

While direct maximization of the likelihood function is non-convex in $w$, the problem can be reformulated as a convex optimization in a new variable: $$v = \frac{w}{\sqrt{\sigma_e^2 \|w\|_2^2 + \sigma_n^2}}$$ yielding the constrained convex program: $$\min_v -\sum_{i=1}^N \log \Phi(y_i h_i^\top v) \quad \text{subject to} \quad \|v\|_2 < 1/\sigma_e$$ Once $v^*$ is obtained, the original parameter is recovered by

$$w = \frac{\sigma_n}{\sqrt{1 - \sigma_e^2 \|v^*\|_2^2}} v^*$$

This reformulation enables:

• Use of standard, efficient convex optimization algorithms (e.g., gradient methods; interior-point methods).
• Strict convexity within the feasible domain, ensuring a unique global minimum.
• More complete analysis of the solution’s uniqueness and likelihood landscape.

5. Theoretical Insights and Validation

Several theoretical findings and empirical results inform practical application:

• The estimator achieves consistency and matches the CRLB asymptotically.
• Simulations demonstrate how performance (MSE) varies with the relative strength of additive and multiplicative noise.
• There exists an "optimal" noise variance—too little noise makes the binary measurement non-informative (all signs are the same); too much destroys information.
• Even the perturbation-ignored estimator offers correct directional information, justifying its use in scenarios focused solely on identifying signal direction.

Simulation results corroborate these findings:

• The ML estimator’s MSE closely approaches the CRLB as sample size increases.
• In small samples, the bias and MSE trade-offs among the full ML, perturbation-ignored, and perfect-sensing estimators are quantitatively illustrated.
• The probability that the unconstrained ML solution satisfies the norm constraint in the convex formulation provides practical conditions for selecting algorithm parameters.

6. Applications, Limitations, and Implications

Sign-based estimation frameworks of this type find application in:

• Binary (1-bit) regression problems, including wireless communications (quantized channel state feedback), robust distributed sensing, and quantized compressed sensing.
• Scenarios with uncertain or fluctuating sensing mechanisms, such as calibration-free or adversarial environments.
• Any setting where only ordinal (sign) information is available, either due to quantization or measurement constraints.

Limitations and Open Questions:

• Estimation of magnitude is fundamentally limited without additive noise.
• Performance degrades with the increase of unmodeled multiplicative perturbation.
• Problem formulation and computational complexity are addressed via convex reparameterization, but high-dimensional regimes may still pose scalability challenges.

Theoretical guarantees and empirical demonstrations collectively establish the robustness of sign-based estimation, its ability to handle structural uncertainty, and its practical utility via efficient convex algorithms. Core findings underscore the importance of accounting for both additive and multiplicative noise, as well as the usefulness of scale-invariant estimation when only the direction of the parameter is consequential.