Topological Parallax: A Geometric Specification for Deep Perception Models
Authors: Abraham Smith, Michael Catanzaro, Gabrielle Angeloro, Nirav Patel, Paul Bendich
NeurIPS 2023 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deeplearning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset. Section 8 illustrates the effectiveness of parallax as a specification, as demonstrated on two models using the cyclo-octane dataset [22]. |
| Researcher Affiliation | Collaboration | Abraham D. Smith Geometric Data Analytics, Inc. 343 W. Main Street Durham, NC 27701 USA abraham.smith@geomdata.com University of Wisconsin-Stout Math, Stats, and CS Dept Menomonie, WI 54751 USA smithabr@uwstout.edu Michael J. Catanzaro Geometric Data Analytics, Inc. 343 W. Main Street Durham, NC 27701 USA michael.catanzaro@geomdata.com Gabrielle Angeloro Geometric Data Analytics, Inc. 343 W. Main Street Durham, NC 27701 USA gabrielle.angeloro@geomdata.com Nirav Patel Geometric Data Analytics, Inc. 343 W. Main Street Durham, NC 27701 USA nirav.patel@geomdata.com Paul Bendich Geometric Data Analytics, Inc. 343 W. Main Street Durham, NC 27701 USA paul.bendich@geomdata.com Duke University Mathematics Dept Durham, NC 27708 USA bendich@math.duke.edu |
| Pseudocode | Yes | Algorithm 7.2 (Estimation of e Pα,0). For each e Rα, sample points p xy along the corresponding line segment xy. (One method of sampling is simply to check the barycenter.) Return True for e if and only if k(p) = 1 p xy. Algorithm 7.5 (Bounding e Pα,ε). For each e R(X, V ), From r = 0, loop: 1. Evaluate k(p) for samples p Dr(e). 2. If k(p) = 0 p, increment r. Else, break. Return the lower bounds p (ρV (e))2 + r2 α and α ρV (e) = ε. |
| Open Source Code | Yes | These algorithms are implemented in Python (and development continues) in our open-source software at https://gitlab.com/geomdata/topological-parallax |
| Open Datasets | Yes | Section 8 illustrates the effectiveness of parallax as a specification, as demonstrated on two models using the cyclo-octane dataset [22]. |
| Dataset Splits | No | For this demonstration, we used a 3-layer fully connected network with a Re LU and a Soft Max, implemented in Py Torch. The network was trained to near-perfect accuracy within a few minutes on a two class problem of real data versus a nearby background. (Hyperparameters and training details are provided in the Supplementary Material.) |
| Hardware Specification | No | These examples may take 100-500 GB of RAM to compute the persistence diagrams as currently implemented. |
| Software Dependencies | Yes | These algorithms are implemented in Python (and development continues) in our open-source software at https://gitlab.com/geomdata/topological-parallax. Because this package relies on GUDHI The GUDHI Project [34], the filtration values are by diameter (not radius). This is important to note, because the theory in the paper is written using radius (not diameter) as the filtration value. The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 3.8.0 edition, 2023. |
| Experiment Setup | No | For this demonstration, we used a 3-layer fully connected network with a Re LU and a Soft Max, implemented in Py Torch. The network was trained to near-perfect accuracy within a few minutes on a two class problem of real data versus a nearby background. (Hyperparameters and training details are provided in the Supplementary Material.) |