Low rank tensor recovery

Reconstructing a tensor from linear measurements

Consider the problem of reconstructing a tensor $T\in\mathbb R^{n_1\times \cdots \times n_d}$ from linear measurements $\mu_1(T), \ldots, \mu_s(T)$, where each $\mu_i: \mathbb R^{n_1\times \cdots \times n_d}\to \mathbb R$ is a linear function.

This problem is called tensor recovery. If the $\mu_i$ are coordinate projections this is also known as tensor completion.

For a fixed measurement $y\in\mathbb R^s$, there are infinitely many tensors $T\in\mathbb R^{n_1\times \cdots \times n_d}$ with

$$\mu(T) = (\mu_1(T), \ldots, \mu_s(T)) = y.$$

Therefore, in order to have the tensor recovery well-posed we need some additional assumption. One common assumption is that the tensor $T$ has low rank. If $s$ is equal to the dimension of rank-$r$ tensors in $\mathbb R^{n_1\times \cdots \times n_d}$, then for a general measurement $\mu$ almost all tensors $T$ can be recovered: $\mu^{-1}(\mu(T))$ is finite. We will confirm this experimentally.

In this example we will recover a $4\times 3\times 2$ tensor of rank $2$ from the following incomplete tensor displayed by two $4\times 3$ slices:

$$ T = \begin{bmatrix} -32 & ? & -24\\ 72 & ? & ? \\ -104 & 40 & ? \\ ? & 16 & 0\end{bmatrix}\quad \begin{bmatrix}? & 10 & ? \\ -57 & 27 & ?\\ -11 & 1 & ?\\ -1 & ? & -7\end{bmatrix}.$$

The symbol ? means that this coordinate is not observed. In julia we use the symbol :? for denoting unobserved entries.

julia> T = reshape([
        -32 :? -24 :? 10 :?;
        72 :? :? -57 27 :?;
        -104 40 :? -11 1 :?;
        :? 16 0 -1 :? -7], 4,3,2)
4×3×2 Array{Any, 3}:
[:, :, 1] =
  -32      :?  -24
   72      :?     :?
 -104    40       :?
     :?  16      0

[:, :, 2] =
    :?  10      :?
 -57    27      :?
 -11     1      :?
  -1      :?  -7

The number of observed entries is 14. This equals he dimension $d$ of rank-$2$ tensors in $\mathbb R^{4\times 3\times 2}$, so recovery is possible, in principle. Let us define the input data for the recovery problem and denote it by $y$:

observed_entries = findall(vec(T) .!= :?)
y = vec(T)[observed_entries]
y = convert(Vector{Float64}, y)

Now, $y$ is a vector that contains the observed coordinates of $T$.

We define the coordinate projection $\mu$ that projects onto the observed coordinates of $T$:

using LinearAlgebra
n = [4; 3; 2]
Id = diagm(0 => ones(prod(n)))
μ = Id[observed_entries, :]

Setting up a polynomial system

To recover $T$ from $\mu(T)$ we define a polynomial system. First, we parametrize rank-2 tensors as follows: $$\mathbb T = a_1 \otimes b_1 \otimes c_1 + a_2 \otimes b_2 \otimes c_2,$$ where $a_i,b_i,c_i\in \mathbb R^3$ for $i=1,2$.

Here, $\mathbb T$ should be understood as a function in $a_1,a_2,b_1,b_2,c_1,c_2$.

To get a one-to-one parametrization we set the last entries of $a_1,a_2,b_1,b_2$ equal to $1$ (this parametrizes a Euclidean dense subset of all rank-$2$ tensors).

Let us implement this in Julia.

using HomotopyContinuation

## 𝕋 = 4x3x2 tensor of rank r=2
r = 2
@var a[1:r, 1:(n[1]-1)] b[1:r, 1:(n[2]-1)] c[1:r, 1:n[3]]
𝕋 = sum( kron(c[i,:], [b[i,:];1], [a[i,:];1]) for i in 1:r)

Now, we can set up a system of polynomials equations for recovery:

F = System(μ * 𝕋 - y, variables = vec([a b c]))

which we solve directly:

julia> S_F = solve(F)
Result with 6 solutions
=======================
• 23 paths tracked
• 6 non-singular solutions (4 real)
• random_seed: 0x40c65a1d
• start_system: :polyhedral

We see that we there are 4 real solutions. They come in pairs of two corresponding to the $\mathbb Z/2\mathbb Z$ action which interchanges the summands in the definition of $T$.

We can check that we get 2 distinct solutions:

recovered_tensors = [evaluate(𝕋, vec([a b c]) => sol) for sol in real_solutions(S_F)]
recovered_tensors = unique_points(recovered_tensors)

The number of recovered tensors is

julia> length(recovered_tensors)
2

Let us take a look at the recovered tensors:

julia> T₁ = reshape(recovered_tensors[1], 4, 3, 2)
4×3×2 Array{Float64, 3}:
[:, :, 1] =
  -32.0       411.518   -24.0
   72.0      1112.91   1075.77
 -104.0        40.0    -728.216
   -1.11448    16.0       0.0

[:, :, 2] =
 -26.0  10.0       -182.054
 -57.0  27.0       -396.574
 -11.0   1.0        -78.642
  -1.0   0.388769    -7.0

The other solutions is

julia> T₂ = reshape(recovered_tensors[2], 4, 3, 2)
4×3×2 Array{Float64, 3}:
[:, :, 1] =
-32.0    8.0  -24.0
 72.0  -40.0  -56.0
-104.0   40.0   -8.0
-40.0   16.0    0.0

[:, :, 2] =
-26.0  10.0   -2.0
-57.0  27.0   21.0
-11.0   1.0  -17.0
-1.0  -1.0   -7.0

It is interesting to observe that both tensors coincide, up to $14$ significant digits, in the unobserved coordinate $(1,1,2)$ with value $-26$. This implies that augmenting $\mu$ with a projection to this coordinate (so it projects to $s = d+1$ coordinates) will not eliminate either completed tensor, even though this number of measurements generically suffices for generic identifiability!

Condition numbers

Let us compute the condition numbers of the recovered tensors. It is defined as

$$\kappa(T) = \frac{1}{\sigma_{\min}(MQ)},$$

where $M$ is the matrix defining $\mu$ and $Q$ is a matrix whose columns form an orthonormal basis for the tangent space of rank-2 tensors at $T$.

We compute $Q$ by using a QR-decomposition of the Jacobian matrix of 𝕋 at the computed solutions.

d = 14
J = differentiate(𝕋, vec([a b c]))
Js = [evaluate(J, vec([a b c]) => sol) for sol in real_solutions(S_F)]
Qs = [(qr(J).Q)[:, 1:d] for J in Js]

We compute the condition numbers for our computed solutions:

julia> [1/svdvals(μ * Q)[end] for Q in Qs]
4-element Vector{Float64}:
 1907.954236908041
   11.641768946763904
 1907.9542369079288
   11.641768946763932

The fact that each number appears twice is again due to the $\mathbb Z/2\mathbb Z$ action on the solutions.

Hence, T₁ has a condition number of about 1907 and T₂ a condition number of about 11. Cite this example:

@Misc{ tensor-recovery2021 ,
    author =  { Paul Breiding, Fulvio Gesmundo, Mateusz Michalek, and Nick Vannieuwenhoven },
    title = { Low rank tensor recovery },
    howpublished = { \url{ https://www.JuliaHomotopyContinuation.org/examples/tensor-recovery/ } },
    note = { Accessed: August 5, 2021 }
}

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