Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.Comment: 18 pages - final version to appear in Theory of Computin

Complexity of linear circuits and geometry

We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border) rigidity, (ii) compute degrees of varieties associated to rigidity, (iii) describe algebraic varieties associated to families of matrices that are expected to have super-linear rigidity, and (iv) prove results about the ideals and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC

A Continual Development Methodology for Large-scale Multitask Dynamic ML Systems

The traditional Machine Learning (ML) methodology requires to fragment the development and experimental process into disconnected iterations whose feedback is used to guide design or tuning choices. This methodology has multiple efficiency and scalability disadvantages, such as leading to spend significant resources into the creation of multiple trial models that do not contribute to the final solution.The presented work is based on the intuition that defining ML models as modular and extensible artefacts allows to introduce a novel ML development methodology enabling the integration of multiple design and evaluation iterations into the continuous enrichment of a single unbounded intelligent system. We define a novel method for the generation of dynamic multitask ML models as a sequence of extensions and generalizations. We first analyze the capabilities of the proposed method by using the standard ML empirical evaluation methodology. Finally, we propose a novel continuous development methodology that allows to dynamically extend a pre-existing multitask large-scale ML system while analyzing the properties of the proposed method extensions. This results in the generation of an ML model capable of jointly solving 124 image classification tasks achieving state of the art quality with improved size and compute cost.Comment: arXiv admin note: text overlap with arXiv:2205.1275

Geometry of Tensors: Open problems and research directions

This is a collection of open problems and research ideas following the presentations and the discussions of the AGATES Kickoff Workshop held at the Institute of Mathematics of the Polish Academy of Sciences (IMPAN) and at the Department of Mathematics of University of Warsaw (MIM UW), September 19-26, 2022.Comment: Comments are welcome. Final version also available at https://agates.mimuw.edu.pl/index.php/research-reports-and-note

Matrix product states and the quantum max-flow/min-cut conjectures

In this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second we generalize this result to a 2d-cycle. In the third we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely a gap of at least 2^{d-2} between the quantum max-flow and the quantum min-cut.Comment: 12 pages, 3 figures - Final version accepted for publication on J. Math. Phy

Moving Through Collective Adversity: Lessons from Posttraumatic Growth Research for Appreciative Inquiry Practice

Tedeschi and Calhoun (1996, 2004) developed the theory of Posttraumatic Growth to explain the experience of growth after trauma. Their work primarily focuses on the individual experience. More recently, Gilpin-Jackson (2014, 2020) and Saul (2014) explored the experience of transformation, healing, and recovery after collective trauma. Appreciative Inquiry was introduced in the late 1980s as a strengths-based approach to organizational change (Cooperrider & Srivastva, 1987). The Appreciative Inquiry method guides an organization through the change process by first selecting the “affirmative topic” to be addressed and then proceeding through a “4-D Cycle” of Discovery, Dream, Design, and Destiny (Cooperrider & Whitney, 2005, p.16). Key findings from the Posttraumatic Growth and collective trauma literature, identifying the factors that enable growth and recovery, can inform and adapt the 4-D Appreciative Inquiry model for use in trauma and adversity contexts. This article argues that the model can be revised to successfully address trauma and adversity through the addition of a new phase or activity: meaning making. The potential for meaning making to create transformative change after collective adversity is demonstrated with examples from the Kalamazoo County Land Bank’s work in Michigan over the last decade

Growth performance, pork quality, and excretion characteristics of pigs fed Bt corn or non-genetically modified corn at two slaughter weights

To compare growth performance and carcass characteristics, 64 pigs (average initial BW = 62 kg) were fed diets containing Bt corn (Bt: Syngenta Bt 11 event) or control corn grain (C: pooled non-biotechnology derived inbred lines). Pigs were blocked by sex and weight and allocated into 16 pens equally. Isocaloric, isonitrogenous diets contained an indigestible marker. Feed and water were provided ad libitum. Feed disappearance and weight gain data, and fecal samples were collected weekly. At slaughter, pigs were divided into two groups based on market weight: 1 (85 kg) and 2 (100 kg). No difference in ADG was observed between pits fed C and Bt diets for harvest weight groups. Feed efficiency was greatest for pigs fed the C diet (P=.002) and was not different between 1 and 2 pigs. An interaction between corn and harvest weight was observed. No corn effects were observed for hot carcass weight; loin eye area; and 1st, 10th, last rib, and last lumbar vertebrae fat. Harvest weight differences were observed; lighter weight pigs had smaller loin eye area (12.4 cm2 vs. 15.7 cm2; P\u3c.0001) and less backfat. Across both market weight groups, gilts had larger LEA (P\u3c.05) and were leaner than barrows. No harvest weight and corn source effects were observed for meat tenderness and drip loss. Hunter color \u27b\u27 values were greater for pigs fed C diets (11.71 vs. 11.31; P=.02) and group 2 pigs (11.77 vs. 11.26; P=.005). There were no significant differences in DM, ether extract, or CP content of meat samples. No corn effects were observed for N or P content of fecal and urine samples. There were no differences in apparent digestibility of N between pigs fed Bt and C diets. Pigs fed C diets had greater apparent P digestibility (57.8% vs. 40.2%; P\u3c.0001). Interactions were observed between treatment and sex; barrows fed Bt corn and gilts fed C corn had lowest apparent P digestibilities. Findings suggest no detrimental effects on growth performance or carcass and excretion characteristics for growing-finishing pigs fed Bt corn. Traits desired by consumers were not different between pigs harvested at light and heavy market weights