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 $d$0 over entries $d$1 (with $d$2), draw $d$3 independent samples $d$4 iid from $d$5, and observe

$d$6

Alternatively, using a differentiable link function $d$7 (logistic or probit), model the conditional probability: $d$8 Typical choices are $d$9 (logistic) and $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$0 (probit).

For fixed $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$1 and rank$$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$2, the number of required 1-bit samples is $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$3 for $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$4, or $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$5 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 $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$6, $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$7,

$$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$8

where $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$9.

• Atomic M-norm:

$r$0

with $r$1 the set of all $r$2-valued rank-1 tensors in $r$3.

For $r$4 and $r$5: $r$6

2.2 Recovery Guarantees

Let $r$7 satisfy $r$8, $r$9. The maximum likelihood estimator with constraints,

$m$0

satisfies with probability $m$1: $m$2 with $m$3 constants depending on $m$4 and $m$5. For $m$6: $m$7 and the error bound becomes $m$8, so $m$9 suffices for fixed $d \geq 2$0 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: $d \geq 2$1 For the logistic model,

$d \geq 2$2

Direct max-qnorm constraints are non-smooth; practical algorithms employ a CP factorization $d \geq 2$3 and projected gradient descent on low-rank factors. Each $d \geq 2$4 is projected so that $d \geq 2$5; projection on $d \geq 2$6 is performed via clipping or line search. Per-iteration cost is $d \geq 2$7 for gradients and $d \geq 2$8 for projections. Empirically, $d \geq 2$9 is effective, with rank tuned via cross-validation. Fast convergence is observed for moderate $N \in \mathbb{N}$0 (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-$N \in \mathbb{N}$1 order-$N \in \mathbb{N}$2 tensor flattened to $N \in \mathbb{N}$3, the matrix rank is at most $N \in \mathbb{N}$4 but the ambient dimension increases exponentially. Sample complexity for matrix completion is therefore $N \in \mathbb{N}$5, compared to $N \in \mathbb{N}$6 for direct tensor methods. For $N \in \mathbb{N}$7 and $N \in \mathbb{N}$8, tensor-based methods are theoretically and empirically more sample-efficient.

Empirical results: On $N \in \mathbb{N}$9 rank-5 tensors with $d$0, max-qnorm tensor completion achieves relative error $d$1, while matricization with nuclear norm minimization yields $d$2 (four times higher). Similar advantages are found for $d$3 tensors (Ghadermarzy et al., 2018).

Method	Relative Error ($d$4 rank-5)	Theoretical Sample Complexity
Max-qnorm Tensor Completion	~0.1	$d$5
Matricization + Matrix Norm	~0.4	$d$6

5. Application: Context-Aware Recommender Systems

In context-aware recommendation, the outcome tensor encodes user-item-context interactions. For users $d$7, items $d$8, and contexts $d$9, form $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$0 with $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$1 the true preference. Observations consist of a sparse set $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$2 of noisy 1-bit samples: $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$3 Fitting proceeds by maximum likelihood with log-loss or hinge loss subject to max-qnorm and supremum constraints. Prediction uses the recovered sign: $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$4 as the inferred user preference.

Empirical performance:

• In-car music data ($$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$5): 1-bit tensor completion achieves $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$6 accuracy (above/below-average), versus $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$7 for matricization.
• Restaurant data ($$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$8): direct tensor method yields $$T \in \mathbb{R}^{N \times N \times \cdots \times N}$$9 sign accuracy, MAE $r$0; improvements of $r$1–$r$2 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 $r$3 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.