Agnostic Estimation for Misspecified Phase Retrieval Models
Authors: Matey Neykov, Zhaoran Wang, Han Liu
NeurIPS 2016 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | In this section we provide numerical experiments based on the three models (2.3), (2.4) and (2.5) where the random variable ε N(0, 1). All models are compared with the Truncated Power Method (TPM), proposed in [37]. For model (2.3) we also compare the results of our approach to the ones given by the TWF algorithm of [7]. Our setup is as follows. In all scenarios the vector β was held fixed at β = ( s 1/2, s 1/2, . . . , s 1/2 | {z } s , 0, . . . 0 | {z } d s ). Since our theory requires that n s2 log d, we have setup four different sample sizes n θs2 log d, where θ {4, 8, 12, 16}. We let s depend on the dimension d and we take s log d. In addition to the suggested approach in Algorithm 1, we also provide results using the refinement procedure (see Algorithm 3.7). We also provide the values of two warm starts of our algorithm, produced by solving program (3.5) with half and full data correspondingly. It is evident that for all scenarios the second step of Algorithms 1 and 2 outperform the warm start from SDP, except in Figure 2 (b), (c), when the sample size is simply two small to for the warm start on half of the data to be accurate. All values we report are based on an average over 100 simulations. The SDP parameter was kept at a constant value (0.015) throughout all simulations, and we observed that varying this parameter had little influence on the final SDP solution. |
| Researcher Affiliation | Academia | Matey Neykov Zhaoran Wang Han Liu Department of Operations Research and Financial Engineering Princeton University, Princeton, NJ 08544 {mneykov, zhaoran, hanliu}@princeton.edu |
| Pseudocode | Yes | Algorithm 1 input :(Yi, Xi)n i=1: data, λn, νn: tuning parameters 1. Split the sample into two approximately equal sets S1, S2, with |S1| = n/2 , |S2| = n/2 . 2. Let bΣ := |S1| 1 P i S1 Yi(X 2 i Id). Solve (3.5). Let bv be the first eigenvector of b A. 3. Let Y = |S2| 1 P i S2 Yi. Solve the following program: bb = argminb(2|S2|) 1P i S2((Yi Y )X i bv X i b)2 + νn b 1. (3.6) 4. Return bβ := bb/ bb 2. |
| Open Source Code | No | No statement or link indicating that the source code for the methodology described in this paper is publicly available was found. |
| Open Datasets | No | In this section we provide numerical experiments based on the three models (2.3), (2.4) and (2.5) where the random variable ε N(0, 1). The paper generates synthetic data; it does not use a pre-existing publicly available dataset. |
| Dataset Splits | Yes | The sample split is required to ensure that after decomposition (3.2), the vector bβ and the value bv β are independent of the remaining sample... To select the νn parameter for (3.6) a pre-specified grid of parameters {ν1, . . . , νl} was selected, and the following heuristic procedure based on K-fold cross-validation was used. We divide S2 into K = 5 approximately equally sized non-intersecting sets S2 = j [K] e Sj 2. For each j [K] and k [l] we run (3.6) on the set r [K],r =j e Sr 2 with a tuning parameter νn = νk to obtain an estimate bβk, e Sj 2. |
| Hardware Specification | No | No specific hardware details (e.g., GPU/CPU models, memory amounts, or cloud resources) used for running experiments were mentioned in the paper. |
| Software Dependencies | No | No specific software dependencies with version numbers (e.g., Python 3.8, PyTorch 1.9, CPLEX 12.4) were mentioned in the paper. |
| Experiment Setup | Yes | Our setup is as follows. In all scenarios the vector β was held fixed at β = ( s 1/2, s 1/2, . . . , s 1/2 | {z } s , 0, . . . 0 | {z } d s ). Since our theory requires that n s2 log d, we have setup four different sample sizes n θs2 log d, where θ {4, 8, 12, 16}. We let s depend on the dimension d and we take s log d...All values we report are based on an average over 100 simulations. The SDP parameter was kept at a constant value (0.015) throughout all simulations...To select the νn parameter for (3.6) a pre-specified grid of parameters {ν1, . . . , νl} was selected, and the following heuristic procedure based on K-fold cross-validation was used. We divide S2 into K = 5 approximately equally sized non-intersecting sets...The TPM is ran for 2000 iterations...the TWF algorithm was also ran at a total number of 2000 iterations, using the tuning parameters originally suggested in [7]. |