Hard Rank: Complexity of Tensor Rank

• Hard rank is the classification of tensor rank as an algebraic feasibility problem, equating its solution to solving polynomial equations over a given domain.
• It establishes a polynomial-time equivalence between the tensor rank decision problem and polynomial equation solving, implying NP-hardness over fields and undecidability over integers.
• Additionally, the framework extends to symmetric tensor rank and minimal rank matrix completion, confirming analogous computational barriers.

Hard rank, in the complexity-theoretic sense established for tensor problems, denotes the classification of tensor rank as an algebraic feasibility problem rather than merely a difficult optimization task. The central result is that, over any integral domain, deciding whether a tensor has rank at most $r$ is polynomial time equivalent to deciding whether a system of polynomial equations has a solution over the same domain. This places tensor rank in exact correspondence with existential polynomial solvability; over $\mathbb{Z}$ the problem is undecidable, over $\mathbb{Q}$ and more generally over fields it is NP-hard, and analogous statements extend to symmetric tensor rank and to minimal rank matrix completion (Shitov, 2016).

1. Tensor rank as an algorithmic invariant

For the purposes of hard rank, the relevant object is the decision problem attached to tensor rank: given a tensor and an integer $r$, determine whether the tensor has rank at most $r$. The decisive contribution of "How hard is the tensor rank?" (Shitov, 2016) is to show that this problem is not merely hard in isolated cases or for specific encodings. Instead, its complexity is characterized exactly in terms of solvability of polynomial systems over the underlying domain.

This perspective changes the status of tensor rank. Rather than treating rank as a numerical invariant whose difficulty depends on ad hoc reductions, the paper identifies tensor rank as a complete encoding device for algebraic feasibility. This suggests that any algorithmic progress for tensor rank would immediately transfer to the corresponding polynomial-equation problem, and conversely.

A plausible implication is that tensor rank should be viewed as a universal algebraic decision problem within this class of domains: its hardness is inherited from the arithmetic of the coefficient domain itself, not from a peculiarity of one tensor format or one reduction.

2. Polynomial-time equivalence with polynomial equation solving

The core technical statement is Theorem 2. Let $R \subset S$ be integral domains, and let $f_1,\ldots,f_p$ be polynomials with coefficients in $R$. There is a polynomial time algorithm that constructs an order-three tensor $T$ over $R$ and an integer $\mathbb{Z}$0 such that the following are equivalent:

1. the equations $\mathbb{Z}$1 have a simultaneous solution in $\mathbb{Z}$2;
2. the rank of $\mathbb{Z}$3 with respect to $\mathbb{Z}$4 does not exceed $\mathbb{Z}$5.

Moreover, these $\mathbb{Z}$6 and $\mathbb{Z}$7 do not depend on the choice of $\mathbb{Z}$8 (Shitov, 2016).

This is sharpened in Theorem 3: for integral domains $\mathbb{Z}$9, deciding whether a tensor with entries in $\mathbb{Q}$0 has rank at most $\mathbb{Q}$1 with respect to $\mathbb{Q}$2 is polynomial time equivalent to deciding whether a given system of polynomial equations with coefficients in $\mathbb{Q}$3 has a solution in $\mathbb{Q}$4 (Shitov, 2016).

The significance of this equivalence is structural. It says that tensor rank is exactly as hard as existential polynomial solving over the same domain. The result is not simply a lower bound such as NP-hardness; it is a bidirectional Turing equivalence. This suggests that tensor rank, in the ordinary nonsymmetric sense, occupies the same complexity class as polynomial-system solvability over the base domain.

3. NP-hardness, undecidability, and domain dependence

The equivalence immediately yields broad complexity consequences. The paper states Corollary 5: tensor rank is NP-hard over any integral domain (Shitov, 2016). The proof route uses the standard fact that solving systems of polynomial equations over an integral domain is NP-hard via reductions from 3-SAT.

A stronger phenomenon appears over the integers. Since Hilbert’s tenth problem is undecidable over $\mathbb{Q}$5, the same transfer principle gives Corollary 4: tensor rank over $\mathbb{Q}$6 is undecidable (Shitov, 2016). This settles a question posed by Gonzalez and Ja'Ja' in 1980. The connection arises because their question, phrased in terms of the multiplicative complexity of simultaneously computing bilinear forms, is equivalent to a tensor-rank question: the rank of the corresponding tensor measures the number of multiplications needed.

The domain dependence is essential. Over $\mathbb{Q}$7, tensor rank inherits undecidability. Over $\mathbb{Q}$8, and more generally over fields, the same framework yields NP-hardness rather than undecidability. This suggests that tensor rank is not a uniformly difficult invariant in one absolute sense; its exact algorithmic status tracks the arithmetic complexity of polynomial solvability over the ambient domain.

4. Symmetric rank and the Hillar–Lim conjecture

The paper generalizes its complexity characterization from ordinary tensor rank to symmetric rank. The symmetric analogue behaves similarly over fields, and the key consequence is Theorem 6: if $\mathbb{Q}$9 is a symmetric tensor, then computing the symmetric rank of $r$0 with respect to any field $r$1 is NP-hard (Shitov, 2016).

This resolves a conjecture of Hillar and Lim. The importance of the result is twofold. First, it shows that the symmetry constraint does not make the computational problem tractable even when the tensor entries are rational. Second, it establishes that the hardness phenomenon is not an artifact of the usual rank notion; it persists in the structured symmetric setting.

A common misconception is that symmetry should simplify rank computation enough to change its qualitative complexity. The result does not support that view. The paper instead shows that symmetric rank admits a computational-complexity description analogous to that of usual tensor rank.

5. Minimal rank matrix completion as a byproduct

A notable byproduct of the construction is a complete characterization of the minimal rank matrix completion problem. Theorem 7 states: let $r$2 be commutative rings. The problem of deciding if a given incomplete matrix with entries in $r$3 has a completion of rank three with respect to $r$4 is polynomial time equivalent to deciding if a given system of polynomial equations with coefficients in $r$5 has a solution over $r$6 (Shitov, 2016).

This result is presented as a complete answer to a question discussed by Buss, Frandsen, and Shallit in 1999. It is also methodologically important because matrix completion serves as an intermediate layer in the proof architecture: polynomial equation solvability is reduced to rank-three matrix completion, and rank-three matrix completion is then reduced to tensor rank.

The consequence is that minimal rank matrix completion shares the same algebraic-feasibility profile as tensor rank. This suggests that these problems are not merely related through heuristic analogies about low-rank structure; they are computationally equivalent in the Turing sense.

6. Conceptual significance of the hard-rank classification

The coherent picture established by the paper is that tensor rank, symmetric tensor rank, and minimal-rank matrix completion all encode the same kind of algebraic feasibility problem: solving polynomial equations over a domain (Shitov, 2016). The importance of this classification lies in its completeness. It does not provide only isolated hardness results, but a domain-sensitive computational description.

In this classification, the central facts are the following. Tensor rank over any integral domain is exactly as hard as solving polynomial equations over that domain. Over $r$7, tensor rank is undecidable. Over $r$8, and more generally over fields, tensor rank is NP-hard. Symmetric rank satisfies an analogous hardness statement, with rational symmetric rank NP-hard. Minimal-rank matrix completion has the same complexity profile.

This suggests a precise meaning for hard rank: rank is hard because it is an exact representation of algebraic solvability. The paper therefore turns tensor rank from a difficult algebraic quantity into a fully classified computational problem, with ordinary rank, symmetric rank, and matrix completion all falling under the same algebraic-complexity template.