A Statistical Model for Tensor Principal Component Analysis
summary
Model: symmetric spiked tensor X = beta v0^{\otimes K} + Z with Gaussian rotation‑invariant noise; the statistical threshold is constant: MLE succeeds once beta > mu_K (e.g. mu3 = 2.870, mu4 = 3.588) but optimizing MLE is intractable. Simple power iteration only works when beta grows polynomially with N (heuristic requirement scales like N^{(K-2)/2}), whereas matricizing (“unfolding”) and doing PCA cuts that exponent roughly in half so practical recovery appears when beta ≳ N^{(K-2)/4} (unfolding reduces exponent to N^{(K-2)/4}); simulations (N≈50–200) match these scalings.
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chapters (4)
- 0:00Start
- 41:59High-dimensional scaling: Power iteration
- 42:31High-dimensional scaling: Unfolding
- 42:53Conclusion
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