Provable Gaussian Embedding with One Observation
Authors: Ming Yu, Zhuoran Yang, Tuo Zhao, Mladen Kolar, Zhaoran Wang
NeurIPS 2018 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | 5 Experiment In this section, we evaluate our methods through experiments. We first justify that although j is unknown, minimizing (3.4) still leads to a consistent estimator. We compare the estimation accuracy with known and unknown covariance matrix j. We set j = σj Toeplitz( j) where Toeplitz( j) denotes Toeplitz matrix with parameter j. We set j U[0, 0.3] and σj U[0.4, 1.6] to make them non-isotropic. The estimation accuracy with known and unknown j are given in Table 1. We can see that although knowing j could give slightly better accuracy, the difference is tiny. Therefore, even if the covariance matrices are not isotropic, ignoring them still gives a consistent estimator. We then consider three kinds of graph structures given in Figure 1: chain structure, !-nearest neighbor structure, and lattice structure. We generate the data according to the conditional distribution (2.3) using Gibbs Sampling. We set p = 100, r = 5 and vary the number of nodes m. For each j, we set j = to be a Toeplitz matrix with i = |i | with = 0.3. We generate independent train, validation, and test sets. For convex relaxation, the regularization parameter is selected using the validation set. We consider two metrics, one is the estimation accuracy kc M M k F /k M k F , and the other is the loss L(c M) on the test set. The simulation results for estimation accuracy for the three graph structures are shown in Figure 2, and the results for loss on test sets are shown in Figure 3. Each result is based on 20 replicates. For the estimation accuracy, we see that when the number of nodes is small, neither method gives accurate estimation; for reasonably large m, non-convex method gives better estimation accuracy since it does not introduce bias; for large enough m, both methods give accurate and similar estimation. For the loss on test sets, we see that in general, both methods give smaller loss as m increases. The non-convex method gives marginally better loss. This demonstrates the effectiveness of our methods. |
| Researcher Affiliation | Academia | Booth School of Business, University of Chicago, Chicago, IL. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. Booth School of Business, University of Chicago, Chicago, IL. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL. |
| Pseudocode | No | The paper describes the algorithms in text, but does not include structured pseudocode or an algorithm block. |
| Open Source Code | No | The paper does not contain any statement about releasing source code or a link to a code repository. |
| Open Datasets | No | We generate the data according to the conditional distribution (2.3) using Gibbs Sampling. |
| Dataset Splits | Yes | We generate independent train, validation, and test sets. For convex relaxation, the regularization parameter is selected using the validation set. |
| Hardware Specification | No | The paper does not provide any specific details about the hardware used for the experiments. |
| Software Dependencies | No | The paper does not mention any specific software dependencies or their version numbers. |
| Experiment Setup | Yes | We set p = 100, r = 5 and vary the number of nodes m. For each j, we set j = to be a Toeplitz matrix with i = |i | with = 0.3. |