Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap beten estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). Hover, the SOS approach does not scale ll to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. #R##N# presents two main contributions. First,  propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result  make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$).  complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.
['Andrea Montanari', 'Nike Sun']
I am writing a research paper on a idea Spectral algorithms for tensor completion
[{"from": "human", "value": "I am writing a research paper on a idea Spectral algorithms for tensor completion"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap beten estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). Hover, the SOS approach does not scale ll to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. #R##N# presents two main contributions. First,  propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result  make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\\ll d^{3/4}$ rather than $r\\ll d^{3/2}$).  complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\\le r \\ll d^{3/2}$ from about $rd^{3/2}$ revealed entries."}]