Tensor Deli: Tensor Completion for Low CP-Rank Tensors via Random Sampling

TL;DR

Two provably accurate methods for low CP-rank tensor completion are proposed - one using adaptive sampling and one using nonadaptive sampling that work well on noisy synthetic data as well as on real world data.

Abstract

We propose two provably accurate methods for low CP-rank tensor completion - one using adaptive sampling and one using nonadaptive sampling. Both of our algorithms combine matrix completion techniques for a small number of slices along with Jennrich's algorithm to learn the factors corresponding to the first two modes, and then solve systems of linear equations to learn the factors corresponding to the remaining modes. For order-$3$ tensors, our algorithms follow a "sandwich" sampling strategy that more densely samples a few outer slices (the bread), and then more sparsely samples additional inner slices (the bbq-braised tofu) for the final completion. For an order-$d$, CP-rank $r$ tensor of size $n \times \cdots \times n$ that satisfies mild assumptions, our adaptive sampling algorithm recovers the CP-decomposition with high probability while using at most $O(nr\log r + dnr)$ samples and $O(n^2r^2+dnr^2)$ operations. Our nonadaptive sampling algorithm recovers the CP-decomposition with high probability while using at most $O(dnr^2\log n + nr\log^2 n)$ samples and runs in polynomial time. Numerical experiments demonstrate that both of our methods work well on noisy synthetic data as well as on real world data.

Figures (8)

• Figure 1: Illustration of the sampling patterns of our adaptive tensor sandwich (left) and nonadaptive tensor sandwich (right) algorithms
• Figure 2: Median relative error (log-scaled) of completed four mode tensors of varying rank as sample complexity increases without noise. Each value is the median of ten trials, $n=100$, $d=4$, $s=2, \gamma \in [0.1,0.8], \delta=8$. We compare the relative errors of adaptive Tensor Deli before and after ten iterations of masked-ALS, as well as just masked ALS alone.
• Figure 3: Median relative error (log-scaled) of completed four mode tensors of varying rank as sample complexity increases without noise. Each value is the median of ten trials, $n=100$, $d=4$, $s=2,\gamma \in [0.1,0.8], \delta=8$. We show errors for both adaptive and non-adaptive TD sampling schemes.
• Figure 4: Comparison of runtime, accuracy and sample complexity for different sized three mode tensors, $n\in[50,120]$, $d=3$, $r=0.1n$. Tensor Deli utilizes KS to complete two densely sampled slices and also performs ten iterations of ALS. This is compared with KS alone used to complete the entire tensor. For this simulated data, Tensor Deli is able to achieve faster runtimes, better accuracy, and at a lower sample complexity than KS. Here we average errors, runtime, and utilized samples over ten independent trials for each choice of $n$.
• Figure 5: Median relative error (log-scaled) of completed three mode tensors of varying rank with additive noise for adaptive Tensor Deli with adaptive sampling method, and KS. Each value is the median of ten trials, for Tensor Deli $n=50$, $d=3$, $s=2, \delta=8$, for KS the proportion of entries that can be sampled for a fiber is 0.7.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 1
• proof
• Lemma 2
• proof