Non-commutative circuits and the sum-of-squares problem.
Bibliographic record
Abstract
We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists an identity (x12+x22+•• + xk2)• (y1^2+y22+•• + yk2)= f12+f22+ ... +fn2, where each fi = fi(X,Y) is bilinear in X={x1,... ,xk} and Y={y1,..., yk}. Over the complex numbers, we show that a sufficiently strong super-linear lower bound on n in, namely, n ≥ k1+e with e >0, implies an exponential lower bound on the size of arithmetic circuits computing the non-commutative permanent.More generally, we consider such sum-of-squares identities for any M polynomial h(X,Y), namely: h(X,Y) = f12+f22+...+fn2.Again, proving n ≥ k1+e in for any explicit h over the complex numbers gives an exponential lower bound for the non-commutative permanent. Our proofs relies on several new structure theorems for non-commutative circuits, as well as a non-commutative analog of Valiant's completeness of the permanent.We proceed to prove such super-linear bounds in some restricted cases. We prove that n ≥ Ω(k6/5) in (1), if f1,..., fn are required to have integer coefficients. Over the real numbers, we construct an explicit M polynomial h such that n in (2) must be at least Ω(k2). Unfortunately, these results do not imply circuit lower bounds. We also present other structural results about non-commutative arithmetic circuits. We show that any non-commutative circuit computing an ordered non-commutative polynomial can be efficiently transformed to a syntactically multilinear circuit computing that polynomial. The permanent, for example, is ordered. Hence, lower bounds on the size of syntactically multilinear circuits computing the permanent imply unrestricted non-commutative lower bounds. We also prove an exponential lower bound on the size of non-commutative syntactically multilinear circuit computing an explicit polynomial. This polynomial is, however, not ordered and an unrestricted circuit lower bound does not follow.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.025 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.018 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".