Mathematics / advanced
Tensor approximation and multilinear structure
Use multilinear algebra to reason about low-rank representations, identifiability, approximation, and computational limits.
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Module 01 advanced
Tensor approximation and multilinear structure
Use multilinear algebra to reason about low-rank representations, identifiability, approximation, and computational limits.
- 01
Tensor products and multilinear maps
Connect coordinate arrays to tensor products, factor multilinear operations through linear maps, and compute separable tensor norms.
35 min - 02
Tensor unfoldings and mode ranks
Choose an explicit unfolding order and distinguish matrix ranks of tensor views from CP rank.
36 min - 03
CP decomposition, border rank, and missing minimizers
Exhibit a real rank-three tensor approached by rank-two tensors and connect diverging components with nonattainment.
41 min - 04
Tucker decomposition and the higher-order SVD
Compress mode subspaces with orthonormal factors, understand core freedom, and compute a concrete HOSVD truncation.
38 min - 05
Tensor trains and matrix product representations
Relate chain bond dimensions to prefix–suffix matrix ranks and track how mode ordering and gauge affect a representation.
40 min - 06
Low-rank approximation and quasi-optimality
Prove a Frobenius tail certificate for HOSVD and compare its guarantee with TT-SVD and exact matrix truncation.
42 min - 07
Identifiability and Kruskal conditions
Check a sufficient CP uniqueness certificate using column independence and distinguish identifiability from numerical stability.
39 min - 08
Alternating least squares and conditioning
Derive a CP factor update with a matching Khatri–Rao order and distinguish blockwise descent from global convergence.
41 min - 09
Tensor completion and sampling
Separate missing-data fitting from recovery guarantees and prove an exact anchored rank-one completion under nonzero observations.
40 min - 10
Tensor networks and contraction complexity
Count dense contraction operations and intermediate storage, showing why a compact network still needs a carefully chosen evaluation order.
39 min