Pure and Spurious Critical Points: a Geometric Study of Linear Networks
Authors: Matthew Trager, Kathlén Kohn, Joan Bruna
ICLR 2020 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Experiment 1. In general, it is very difficult to describe the open regions in Rn that are separated by the ED discriminant of a variety V Rn. Finding the typical number of real critical points for the distance function hu restricted to V, requires the computation of the volumes of these open regions. In the current state of the art in real algebraic geometry, this is only possible for very particular varieties V. For these reasons, and to get more insights on the typical number of real critical points of determinantal varieties with a perturbed Euclidean distance, we performed computational experiments with Macaulay2 (Grayson & Stillman, 2019) in the situation of Example 13. We fixed the determinantal variety M1 R3 3 of rank-one (3 3)-matrices. In each iteration of the experiment, we picked a random automorphism ϕ : R3 3 R3 3 and a random matrix Q0 R3 3. We first verified that the number of complex critical points of the perturbed quadratic distance function hϕ,Q0 restricted to M1 is the expected number 39. After that, we computed the number of real critical points and the number of local minima among them. Our results for 2000 iterations can be found in Table 2 and Figure 3. |
| Researcher Affiliation | Academia | Matthew Trager New York University Kathl en Kohn KTH Stockholm Joan Bruna New York University |
| Pseudocode | No | The paper does not contain any pseudocode or algorithm blocks. It references using the Macaulay2 software system for its computational experiments. |
| Open Source Code | No | The paper does not provide or link to any open-source code for the methodology or analysis described. It mentions using Macaulay2, which is third-party software, but no custom code is shared. |
| Open Datasets | No | The paper describes data generation for its computational experiments as: 'The entries of the random matrix Q0 are independently and uniformly chosen among the integers in Z = { 10, 9, . . . , 9, 10}. The random automorphism ϕ is given by a matrix in Z9 9 whose entries are also chosen independently and uniformly at random.' This is a description of how synthetic data was generated for the experiments, not a reference to a publicly available or open dataset with access information (link, DOI, specific citation). |
| Dataset Splits | No | The paper describes data generation for its computational experiments rather than using a traditional dataset with predefined splits. It does not provide any specific information about training, validation, or test dataset splits. |
| Hardware Specification | No | The paper does not explicitly describe the hardware used to run its computational experiments. |
| Software Dependencies | No | The paper mentions 'Macaulay2 (Grayson & Stillman, 2019)' as software used for computational experiments. However, it does not specify a version number for Macaulay2, which is required for a reproducible description of ancillary software. |
| Experiment Setup | Yes | The paper provides details for its computational experiments under 'Implementation details': 'The entries of the random matrix Q0 are independently and uniformly chosen among the integers in Z = { 10, 9, . . . , 9, 10}. The random automorphism ϕ is given by a matrix in Z9 9 whose entries are also chosen independently and uniformly at random.' This describes the parameters and method for generating the inputs for their experiments. |