On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology Article Swipe
Suryaka Suresh
,
Bishshoy Das
,
Vinayak Abrol
,
Sumantra Dutta Roy
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2311.04592
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2311.04592
We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. Motivated by existing studies using simplicial complexes on shallow fully connected networks (FCN), we present an extended analysis using Cubical homology instead, with a variety of popular deep architectures and real image datasets. We demonstrate that as depth increases, a topologically complicated dataset is transformed into a simple one, resulting in Betti numbers attaining their lowest possible value. The rate of decay in topological complexity (as a metric) helps quantify the impact of architectural choices on the generalization ability. Interestingly from a representation learning perspective, we highlight several invariances such as topological invariance of (1) an architecture on similar datasets; (2) embedding space of a dataset for architectures of variable depth; (3) embedding space to input resolution/size, and (4) data sub-sampling. In order to further demonstrate the link between expressivity \& the generalization capability of a network, we consider the task of ranking pre-trained models for downstream classification task (transfer learning). Compared to existing approaches, the proposed metric has a better correlation to the actually achievable accuracy via fine-tuning the pre-trained model.
Related Topics
Concepts
Embedding
Betti number
Persistent homology
Topology (electrical circuits)
Artificial neural network
Generalization
Metric (unit)
Computer science
Mathematics
Artificial intelligence
Feature (linguistics)
Theoretical computer science
Pattern recognition (psychology)
Algorithm
Discrete mathematics
Combinatorics
Economics
Mathematical analysis
Philosophy
Operations management
Linguistics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2311.04592
- https://arxiv.org/pdf/2311.04592
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4388585318
All OpenAlex metadata
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https://openalex.org/W4388585318Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2311.04592Digital Object Identifier
- Title
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On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic TopologyWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2023Year of publication
- Publication date
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2023-11-08Full publication date if available
- Authors
-
Suryaka Suresh, Bishshoy Das, Vinayak Abrol, Sumantra Dutta RoyList of authors in order
- Landing page
-
https://arxiv.org/abs/2311.04592Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2311.04592Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2311.04592Direct OA link when available
- Concepts
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Embedding, Betti number, Persistent homology, Topology (electrical circuits), Artificial neural network, Generalization, Metric (unit), Computer science, Mathematics, Artificial intelligence, Feature (linguistics), Theoretical computer science, Pattern recognition (psychology), Algorithm, Discrete mathematics, Combinatorics, Economics, Mathematical analysis, Philosophy, Operations management, LinguisticsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.passes | 12 |
| abstract_inverted_index.simple | 73 |
| abstract_inverted_index.value. | 83 |
| abstract_inverted_index.Cubical | 45 |
| abstract_inverted_index.between | 154 |
| abstract_inverted_index.changes | 9 |
| abstract_inverted_index.choices | 100 |
| abstract_inverted_index.dataset | 68, 131 |
| abstract_inverted_index.feature | 6 |
| abstract_inverted_index.further | 150 |
| abstract_inverted_index.metric) | 93 |
| abstract_inverted_index.network | 21 |
| abstract_inverted_index.numbers | 78 |
| abstract_inverted_index.popular | 52 |
| abstract_inverted_index.present | 40 |
| abstract_inverted_index.ranking | 168 |
| abstract_inverted_index.several | 113 |
| abstract_inverted_index.shallow | 34 |
| abstract_inverted_index.similar | 124 |
| abstract_inverted_index.studies | 29 |
| abstract_inverted_index.through | 13, 23 |
| abstract_inverted_index.variety | 50 |
| abstract_inverted_index.Compared | 177 |
| abstract_inverted_index.ability. | 104 |
| abstract_inverted_index.accuracy | 192 |
| abstract_inverted_index.actually | 190 |
| abstract_inverted_index.analysis | 43 |
| abstract_inverted_index.consider | 164 |
| abstract_inverted_index.existing | 28, 179 |
| abstract_inverted_index.extended | 42 |
| abstract_inverted_index.homology | 46 |
| abstract_inverted_index.instead, | 47 |
| abstract_inverted_index.learning | 109 |
| abstract_inverted_index.network, | 162 |
| abstract_inverted_index.networks | 37 |
| abstract_inverted_index.numbers. | 25 |
| abstract_inverted_index.possible | 82 |
| abstract_inverted_index.proposed | 182 |
| abstract_inverted_index.quantify | 95 |
| abstract_inverted_index.topology | 4 |
| abstract_inverted_index.variable | 135 |
| abstract_inverted_index.(transfer | 175 |
| abstract_inverted_index.Motivated | 26 |
| abstract_inverted_index.attaining | 79 |
| abstract_inverted_index.complexes | 32 |
| abstract_inverted_index.connected | 36 |
| abstract_inverted_index.datasets. | 58 |
| abstract_inverted_index.datasets; | 125 |
| abstract_inverted_index.embedding | 7, 127, 138 |
| abstract_inverted_index.highlight | 112 |
| abstract_inverted_index.resulting | 75 |
| abstract_inverted_index.achievable | 191 |
| abstract_inverted_index.capability | 159 |
| abstract_inverted_index.complexity | 90 |
| abstract_inverted_index.downstream | 172 |
| abstract_inverted_index.increases, | 64 |
| abstract_inverted_index.invariance | 118 |
| abstract_inverted_index.learning). | 176 |
| abstract_inverted_index.simplicial | 31 |
| abstract_inverted_index.approaches, | 180 |
| abstract_inverted_index.complicated | 67 |
| abstract_inverted_index.correlation | 187 |
| abstract_inverted_index.demonstrate | 60, 151 |
| abstract_inverted_index.fine-tuning | 194 |
| abstract_inverted_index.invariances | 114 |
| abstract_inverted_index.pre-trained | 169, 196 |
| abstract_inverted_index.topological | 89, 117 |
| abstract_inverted_index.transformed | 70 |
| abstract_inverted_index.architecture | 122 |
| abstract_inverted_index.expressivity | 155 |
| abstract_inverted_index.perspective, | 110 |
| abstract_inverted_index.well-trained | 18 |
| abstract_inverted_index.Interestingly | 105 |
| abstract_inverted_index.architectural | 99 |
| abstract_inverted_index.architectures | 54, 133 |
| abstract_inverted_index.sub-sampling. | 146 |
| abstract_inverted_index.topologically | 66 |
| abstract_inverted_index.classification | 173 |
| abstract_inverted_index.generalization | 103, 158 |
| abstract_inverted_index.representation | 108 |
| abstract_inverted_index.resolution/size, | 142 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 4 |
| sustainable_development_goals[0].id | https://metadata.un.org/sdg/11 |
| sustainable_development_goals[0].score | 0.7599999904632568 |
| sustainable_development_goals[0].display_name | Sustainable cities and communities |
| citation_normalized_percentile |