Bibliographic record
Abstract
An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in $$\mathbb{F}[x_1, x_2, \ldots , x_n]$$ . An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial $$x^n_1 + x^n_2 + \cdots + x^n_n$$ has at least $$\Omega(n^2)$$ vertices (and hence edges). To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of $$\Omega(n \,{\rm log}\, n)$$ on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (Numer Math 20:238–251, 1973) and Baur and Strassen (Theor Comput Sci 22:317–330, 1983). On the way, we also get an alternate and unified proof of an $$\Omega(n \,{\rm log}\, n)$$ lower bound on the size of a homogeneous arithmetic circuit (follows from the work of Strassen (1973) and Baur & Strassen (1983)), and an n/2 lower bound $$(n \,{\rm over}\, \mathbb{R})$$ on the determinantal complexity of an explicit polynomial (Mignon and Ressayre in Int Math Res Notes 2004(79):4241–4253, 2004; Cai et al. in Comput Complex 19(1):37–56, 2010, http://dx.doi.org/10.1007/s00037-009-0284-2 ; Yabe in CoRR, 2015, http://arxiv.org/abs/1504.00151 ). These are currently the best lower bounds known for these problems for any explicit polynomial and were originally proved nearly two decades apart using seemingly different proof techniques.
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 | 0.004 | 0.025 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.005 | 0.016 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.002 | 0.010 |
| Insufficient payload (model declined to judge) | 0.038 | 0.007 |
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".