Authors: Matteo Bechere, Henri Breloer
Tensors with symmetries
Tensor with structured decomposition
Tensors are a way of representing multidimensional information and they appear in fields ranging from engineering and statistics, to signal processing and scientific computing. Unfortunately, tensors can become large and difficult to interpret. Tensor decomposition addresses this problem by expressing a complicated tensor as a sum of simpler building blocks, known as simple tensors. Decompositions often reveal patterns which are hidden in the original data. They also provide a compact representation: the smallest number of simple building blocks needed to reconstruct a tensor is called its rank and it is a measure of the tensor’s complexity.
Where structure enters
Many tensors encountered in practice are not arbitrary. Their entries may be linked to symmetries or constraints, so the simple components in their decomposition are expected to lie on a particular geometric shape.
In this recent work, we study symmetric tensors whose components lie on a rational variety. The points of this geometrical shape can be generated by a small collection of parameters using polynomial or rational formulae. Instead of searching freely through the whole space, we know in advance that the building blocks must lie in this lower-dimensional shape.
Turning structure into an advantage
This additional constraint might appear to make the (already hard) decomposition problem harder! In fact, it can make the problem substantially smaller. The central idea is to translate the original structured tensor into another symmetric tensor involving fewer variables, though generally of higher order. Under suitable assumptions, decomposing this smaller object is equivalent to decomposing the original one (Theorem 3.17.). The correspondence preserves the number of components: every decomposition of the reduced tensor produces a structured decomposition of the original tensor with the same length, and vice versa.
The resulting strategy (Algorithm 4.1.) is therefore:
- Recognize the tensor’s geometric structure.
- Transform the tensor into a lower-dimensional representation.
- Decompose the smaller representation.
- Map the result back to the original tensor.
This is valuable because standard decomposition methods, that don’t exploit structured data, may struggle with the original tensor, even when a short decomposition exists. In Symmetric tensor decomposition on rational varieties we give an example (Section 4.1) in which a direct method fails on a symmetric tensor of size 6 and order 4, while the structure-aware approach makes the decomposition possible.
The takeaway message is that having structure on your tensors is not just an extra restriction, instead it is useful geometric information that, when incorporated in the decomposition methods, often allows to solve decomposition problems that would otherwise be out of reach.
To dive deeper into the topic, more technical presentation of parts of the paper Symmetric tensor decomposition on rational varieties can be found on Youtube.
