Low-rank approximation and quasi-optimality

For each mode n, let Pₙ project onto the leading rₙ left singular vectors of its original unfolding. Let τₙ be the Frobenius norm of the discarded singular-value tail. The HOSVD approximation applies all these mode projections. Telescope its residual by projecting modes one after another. Because different mode projections commute and are orthogonal, the successive lost pieces are mutually orthogonal; each piece has norm at most the corresponding unprojected mode residual τₙ.

Pythagoras therefore bounds the squared total error by the sum of τₙ squared. The same original unfolding yields a lower bound on every feasible multilinear-rank approximation: its mode-n matrix rank is at most rₙ, so the matrix Eckart–Young theorem forces error at least τₙ. Taking the largest tail gives a computable lower certificate. No tensor optimization algorithm is needed to establish either inequality.

The rank-bounded Tucker set is closed because each unfolding rank condition is described by vanishing matrix minors. A minimizing sequence is bounded near the fixed target tensor and has a convergent subsequence in finite dimension, proving attainment. This differs from the nonclosed CP-rank set. The HOSVD bound compares with that attained minimum, but nothing in its independent mode choices jointly optimizes the selected subspaces.

For an order-three tensor with entries at (1, 1, 1), (1, 2, 2), and (2, 1, 2), all equal to one, the leading directions in modes one and two are e₁, while mode three chooses e₂. Their product coordinate (1, 1, 2) has value zero, so the rank-(1, 1, 1) HOSVD approximation is zero. Keeping just the first nonzero coordinate already gives a smaller error. This counterexample has no leading singular-value ties, so the failure is not a tie-breaking accident.

TT-SVD performs sequential SVDs of reshaped remainders and controls discarded errors through orthogonal factors. For fixed TT bond bounds on a d-way tensor with d ≥ 2, the Frobenius error is bounded by the square root of the sum of squared optimal discarded tails of its d − 1 original prefix–suffix unfoldings. Each such tail is at most the best error under those TT bounds, giving a √(d − 1) quasi-optimal factor.

The original-cut tail bounds are certificates; the actual sequential matrices change as earlier projections are applied. To budget a total absolute tolerance ε, it is sufficient to make each of the d − 1 local discarded Frobenius tails at most ε/√(d − 1). This is a bound on an entire tail norm, not a threshold on each individual singular value. Comparing HOSVD and TT guarantees still requires their different rank constraints and mode ordering, rather than choosing whichever constant looks smaller.