Geometric structure of graph Laplacian embeddings
Authors: Nicolás García Trillos, Franca Hoffmann, Bamdad Hosseini
JMLR 2021 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We then generate a random graph by drawing n = 211 vertices from ρ and construct the graph Laplacian matrix n as in (5) by setting ε = 2(log n)3/4 n1/2 0.2. We then construct the discrete Laplacian embedding Fn using the first two eigenvectors of n and compute the number noutside of embedded points that fall outside the cones... We choose |γ| {0.7, 0.8, 0.9, 1, 1.1, 1.2} and repeat the experiment over 20 trials for each value of |γ| where the vertices are redrawn from the mixture. We report the averaged values of noutside/n, as an empirical approximation to the probability mass outside the cones, versus |γ|2 in Figure 2(a) indicating that indeed log(noutside/n) = O(|γ|2)) as expected. |
| Researcher Affiliation | Academia | Nicol as Garc ıa Trillos EMAIL Department of Statistics University of Wisconsin-Madison Madison, WI 53706, USA Franca Hoffmann EMAIL Bamdad Hosseini EMAIL Computing and Mathematical Sciences California Institute of Technology Pasadena, CA 91125, USA |
| Pseudocode | No | The paper provides mathematical definitions, theorems, and proofs, but does not include any clearly labeled pseudocode or algorithm blocks. |
| Open Source Code | No | The paper does not contain any explicit statements or links indicating that source code for the methodology is openly available. |
| Open Datasets | No | The paper describes generating synthetic data from defined distributions (e.g., 'mixture of two standard Gaussian densities', 'probability measure ρ on a dumbbell shaped domain'). It does not provide concrete access information (links, DOIs, repositories) for pre-existing or specifically generated public datasets. |
| Dataset Splits | No | The paper's numerical experiments involve drawing random samples (e.g., 'n = 211 vertices from ρ') and repeating trials, but it does not describe dataset splits for training, validation, or testing. |
| Hardware Specification | No | The paper does not specify any hardware details (e.g., GPU/CPU models, memory) used for running its experiments or simulations. |
| Software Dependencies | No | The paper does not mention any specific software packages, libraries, or their version numbers used in the experimental setup. |
| Experiment Setup | Yes | We then generate a random graph by drawing n = 211 vertices from ρ and construct the graph Laplacian matrix n as in (5) by setting ε = 2(log n)3/4 n1/2 0.2. We then construct the discrete Laplacian embedding Fn using the first two eigenvectors of n and compute the number noutside of embedded points that fall outside the cones C((1, 1)T , π/4 1/4, 0) and C((1, 1)T , π/4 1/4, 0). We choose |γ| {0.7, 0.8, 0.9, 1, 1.1, 1.2} and repeat the experiment over 20 trials for each value of |γ| where the vertices are redrawn from the mixture. ... We let ℓ= 1 and consider dumbbell shaped domains with parameter ϑ {.05, .1, .15, .2, .25, .3, .4, .5}. We then let ρ denote the uniform measure on the dumbbell shaped domain M and generate random graphs with n = 211 vertices sampled randomly from ρ. We choose the rest of the parameters in this experiment identical to the experiment in Example 1 and report the averaged values of noutside/n versus ϑ over 20 trials. |