5.7.2 Tensor Simple Factor Separation

Tensor Simple Factor Separation is the process, and the associated notion, of recovering the individual factor vectors that compose a simple (pure, decomposable) tensor from the tensor itself, so that the single combined object is separated back into the distinct vectors — one per participating vector space — whose tensor product produced it.

Formal Setting

Let V1, V2, …, Vn be vector spaces over a field F, and let t be a simple tensor in V1 ⊗ V2 ⊗ ⋯ ⊗ Vn, meaning there exist vectors v1, …, vn with

Factor separation is the operation that, given t (or its coordinate array), outputs a tuple (v1, v2, …, vn) reproducing this equation. The operation is only defined on the subset of simple tensors, since a tensor that is not simple has no single tuple of factors to output.

The Two-Factor Case

The clearest and most computationally direct setting for factor separation is the two-factor case, where the tensor product reduces to matrices.

Matrices and Rank-One Separation

When t lies in V1 ⊗ V2 with V1 and V2 finite-dimensional, choosing bases identifies t with a matrix M. The tensor is simple exactly when M has rank at most one, and in that case M can be separated as

for column vectors a and b, where a supplies the coordinates of v1 and b supplies the coordinates of v2. Separation here can be carried out explicitly: any nonzero row of M is proportional to b, and any nonzero column of M is proportional to a, so a single nonzero entry together with its row and column already determines the separation up to the standard scalar rebalancing ambiguity.

Singular Value Decomposition as a Separation Tool

For a rank-one matrix, the singular value decomposition collapses to a single term, M = σ u vᵀ, and the unit vectors u and v obtained from this decomposition give a normalized factor separation, with the singular value σ absorbing the scale that would otherwise be split ambiguously between the two factors.

Separation for More Than Two Factors

Beyond two factors, separation becomes more delicate because there is no single matrix rank criterion that applies uniformly, and different unfoldings of the tensor must be examined together.

Flattening and Rank-One Conditions

A common approach flattens (matricizes) the n-way tensor along each factor in turn, producing n different matrices. The tensor is simple exactly when every one of these flattenings has rank at most one, and the leading singular vector of the i-th flattening recovers the i-th factor vi up to scale. Consistency across all n flattenings is what confirms that a genuine simple-tensor separation exists, rather than merely a rank-one approximation along one direction.

Iterative and Alternating Methods

When separation is attempted numerically on a tensor that is only approximately simple, alternating methods fix all factors but one, solve a linear least-squares problem for the remaining factor, and cycle through the factors repeatedly. For a tensor that is exactly simple, such methods converge to the true separation in a single pass per factor; for near-simple tensors they converge to a best rank-one approximation, which functions as an approximate separation.

Ambiguity Inherent to Separation

Because the same simple tensor can be written from more than one factor list, factor separation never returns a single canonical answer without an additional normalization convention.

Scalar Redistribution

If (v1, …, vn) is a valid separation of t, then so is (λ1 v1, …, λn vn) for any scalars satisfying λ1 λ2 ⋯ λn = 1. Any separation procedure must therefore either accept this multi-parameter ambiguity as intrinsic to the problem or fix a convention — such as normalizing every factor except one to unit length — to select a single representative from the ambiguity class.

Sign and Phase Ambiguity

Even after norm normalization, an overall sign (over the real numbers) or phase (over the complex numbers) can still be redistributed among the factors while satisfying the product constraint, so a fully canonical separation additionally requires fixing a sign or phase convention, for instance by requiring the first nonzero coordinate of each factor to be positive.

Obstruction to Separation

Not every tensor admits a separation, and recognizing this obstruction is as important as performing the separation itself when it exists.

Non-Simple Tensors

A tensor with rank greater than one has no valid separation into a single factor tuple; attempts to separate such a tensor by any single-pass method necessarily fail to reproduce it exactly; only an approximate, best-fit separation is available, and it captures at most the dominant rank-one component of the tensor.

Detecting the Obstruction

The standard way to detect whether separation is even possible is to test the rank-one conditions described above — vanishing of all appropriate flattenings' higher-order minors — before attempting separation, so that a numerical method is not misapplied to a tensor for which no exact separation exists.

Illustrative Diagram

The diagram shows the single combined tensor on the left being separated into its individual factor vectors on the right, the inverse operation of the tensor product that originally combined them.