1-Bit Tensor Completion: Theory & Applications

• 1-bit tensor completion is the process of estimating a low-rank, high-order tensor from sparse binary (±1) measurements, generalizing matrix completion.
• It employs convex relaxations via max-qnorm and atomic M-norm constraints to achieve near-optimal sample complexity and robust recovery.
• The method is effectively applied in context-aware recommender systems, improving prediction accuracy over traditional matricization approaches.

1-bit tensor completion is the problem of efficiently estimating a low-rank, high-order tensor from partial binary (±1) measurements of its entries. This setting generalizes the well-studied 1-bit matrix completion problem to tensors of order $d \geq 2$. The goal is to reconstruct an order-$d$ tensor $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$ of CP-rank $r$, given only $m$ noisy, quantized (1-bit) samples of selected entries. Regularization via the max-qnorm or atomic M-norm enables tractable convex relaxations matching the sample complexity of unquantized tensor completion. Applications include context-aware recommender systems, where observations typically consist of binary user preferences or implicit feedback.

1. Formal Problem Statement

Let $d \geq 2$, $N \in \mathbb{N}$, and an unknown order-$d$ tensor $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$ of rank at most $r$. Observations are produced according to a stochastic measurement model: Given a sampling distribution $\Pi = \{\pi_\omega\}$ over entries $\omega = (i_1, ..., i_d) \in [N]^d$ (with $\sum_\omega \pi_\omega = 1$), draw $m$ independent samples $\Omega = \{\omega_1, ..., \omega_m\}$ iid from $\Pi$, and observe

$$y_\omega = \operatorname{sign}(T_\omega + \zeta_\omega) \in \{\pm 1\}, \quad \zeta_\omega \sim \text{noise}.$$

Alternatively, using a differentiable link function $f : \mathbb{R} \to [0,1]$ (logistic or probit), model the conditional probability: $$P(y_\omega = +1|T_\omega) = f(T_\omega), \qquad P(y_\omega = -1|T_\omega) = 1 - f(T_\omega).$$ Typical choices are $f(x) = e^x/(1+e^x)$ (logistic) and $f(x) = \Phi(x/\sigma)$ (probit).

For fixed $\|\mathbf{T}\|_\infty \leq \alpha$ and rank$(T) \leq r$, the number of required 1-bit samples is $m = O(Nd)$ for $r = O(1)$, or $m = O(r^{O(d^2)} N d)$ in general, up to a specified reconstruction accuracy (Ghadermarzy et al., 2018).

2. Theoretical Foundations

2.1 Max-qnorm and Atomic M-norm Regularizers

Direct rank constraints are non-convex in the tensor setting. Two tractable surrogates are employed:

• Max-qnorm: For $T = U^{(1)} \odot \cdots \odot U^{(d)}$, $U^{(j)} \in \mathbb{R}^{N \times R}$,

$$\|T\|_{\text{max--}q} = \min \prod_{j=1}^d \|U^{(j)}\|_{2,\infty},$$

where $\|U\|_{2,\infty} = \max_i \| \text{row}_i(U) \|_2$.

• Atomic M-norm:

$$\|T\|_M = \inf \left\{ t > 0 : T \in t \cdot \operatorname{conv}(\mathcal{A}) \right\}$$

with $\mathcal{A}$ the set of all $\pm 1$-valued rank-1 tensors in $\{\pm 1\}^{N^d}$.

For $\operatorname{rank}(T) = r$ and $\|T\|_\infty \leq \alpha$: $$\alpha \leq \|T\|_M \leq (r\sqrt{r})^{d-1} \alpha, \qquad \alpha \leq \|T\|_{\text{max--}q} \leq \sqrt{r^{d^2 - d}} \alpha.$$

2.2 Recovery Guarantees

Let $T^\star$ satisfy $\|T^\star\|_\infty \leq \alpha$, $\|T^\star\|_{\text{max--}q} \leq R_{\max}$. The maximum likelihood estimator with constraints,

$$\hat{T}_{\max} = \arg\min_{\|X\|_\infty \leq \alpha,\, \|X\|_{\text{max--}q} \leq R_{\max}} \sum_{\omega \in \Omega} \left[ 1_{y_\omega = +1} (-\log f(X_\omega)) + 1_{y_\omega = -1} (-\log (1-f(X_\omega))) \right]$$

satisfies with probability $\geq 1 - \delta$: $$\|T^\star - \hat{T}_{\max}\|_\Pi^2 \leq C_{\max} c_2^d \beta_\alpha \left( L_\alpha R_{\max} \sqrt{\frac{dN}{m}} + U_\alpha \sqrt{ \frac{\log(4/\delta)}{m}} \right)$$ with $L_\alpha, \beta_\alpha, U_\alpha$ constants depending on $f$ and $\alpha$. For $\operatorname{rank}(T^\star) = r$: $R_{\max} = \sqrt{r^{d^2-d}} \alpha$ and the error bound becomes $O(\sqrt{dN/m} r^{O(d^2)})$, so $m = O(Nd)$ suffices for fixed $r$ and error (Ghadermarzy et al., 2018).

The proof relies on bounding the Rademacher complexity of the feasible class and using Bernstein/Hoeffding concentration to guarantee that low negative log-likelihood implies low squared error.

3. Algorithmic Approaches

A convex program is solved for maximum likelihood estimation under either max-qnorm or M-norm (and supremum) constraints: $$\min_{X \in \mathbb{R}^{N^d}} \mathcal{L}(X;y), \quad \text{subject to} \quad \|X\|_{\text{max--}q} \leq C_1,\, \|X\|_M \leq C_2,\, \|X\|_\infty \leq \alpha.$$ For the logistic model,

$$\mathcal{L}(X;y) = \sum_{\omega \in \Omega} \left[ 1_{y_\omega=+1}(-\log \sigma(X_\omega)) + 1_{y_\omega=-1}(-\log (1-\sigma(X_\omega))) \right], \quad \sigma(x) = \frac{1}{1 + e^{-x}}.$$

Direct max-qnorm constraints are non-smooth; practical algorithms employ a CP factorization $X = V^{(1)} \odot \cdots \odot V^{(d)}$ and projected gradient descent on low-rank factors. Each $V^{(j)}$ is projected so that $\max_j \|V^{(j)}\|_{2,\infty} \leq C_1^{1/d}$; projection on $\|\cdot\|_\infty$ is performed via clipping or line search. Per-iteration cost is $O(m R d)$ for gradients and $O(N R d)$ for projections. Empirically, $R \approx 2N$ is effective, with rank tuned via cross-validation. Fast convergence is observed for moderate $N$ (hundreds per mode) (Ghadermarzy et al., 2018).

4. Comparison to Matricization Approaches

A competing heuristic is matricization: flattening the tensor into a matrix (e.g., splitting modes in half) and applying 1-bit matrix completion protocols. For a rank-$r$ order-$d$ tensor flattened to $N^{d/2} \times N^{d/2}$, the matrix rank is at most $r$ but the ambient dimension increases exponentially. Sample complexity for matrix completion is therefore $m = O(r N^{d/2} \log N)$, compared to $O(r^{O(d^2)} N d)$ for direct tensor methods. For $d \geq 3$ and $r \ll N^{d/2-1}$, tensor-based methods are theoretically and empirically more sample-efficient.

Empirical results: On $30 \times 30 \times 30$ rank-5 tensors with $m/N^3 \approx 0.3$, max-qnorm tensor completion achieves relative error $\sim 0.1$, while matricization with nuclear norm minimization yields $\sim 0.4$ (four times higher). Similar advantages are found for $15^4$ tensors (Ghadermarzy et al., 2018).

Method	Relative Error ($30^3$ rank-5)	Theoretical Sample Complexity
Max-qnorm Tensor Completion	~0.1	$O(r^{O(d^2)} N d)$
Matricization + Matrix Norm	~0.4	$O(r N^{d/2} \log N)$

5. Application: Context-Aware Recommender Systems

In context-aware recommendation, the outcome tensor encodes user-item-context interactions. For users $U$, items $I$, and contexts $C$, form $T^\star \in \mathbb{R}^{|U| \times |I| \times |C|}$ with $T^\star_{u,i,c}$ the true preference. Observations consist of a sparse set $\Omega$ of noisy 1-bit samples: $$y_{u,i,c} = \operatorname{sign}(T^\star_{u,i,c} + \zeta_{u,i,c}).$$ Fitting proceeds by maximum likelihood with log-loss or hinge loss subject to max-qnorm and supremum constraints. Prediction uses the recovered sign: $\operatorname{sign}(X_{u,i,c})$ as the inferred user preference.

Empirical performance:

• In-car music data ($42 \times 140 \times 26$): 1-bit tensor completion achieves $\sim 77\%$ accuracy (above/below-average), versus $\sim 60\%$ for matricization.
• Restaurant data ($40 \times 50 \times 6$): direct tensor method yields $84\%$ sign accuracy, MAE $\sim 0.76$; improvements of $15$–$20\%$ over both context-free and flattened matrix baselines (Ghadermarzy et al., 2018).

6. Summary and Implications

1-bit tensor completion by constrained likelihood (max-qnorm or atomic M-norm) achieves sample complexity $O(Nd)$ for fixed-rank tensors. This matches the information-theoretic rates of unquantized measurements and outperforms matricization, both theoretically and empirically, for high-order tensors. This methodology provides a robust foundation for learning in settings with only coarse, high-dimensional, and binary feedback, exemplified by context-aware recommender systems (Ghadermarzy et al., 2018). A plausible implication is that for structured high-dimensional problems with severe quantization or limited feedback, tensor-based approaches with suitable convex constraints should be preferred over methods relying on data flattening.