Some Remarks on Diffusion Distances
As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the 𝐿2 distance between two 𝐿2-normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast t...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2010/464815 |
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| author | Maxim J. Goldberg Seonja Kim |
| author_facet | Maxim J. Goldberg Seonja Kim |
| author_sort | Maxim J. Goldberg |
| collection | DOAJ |
| description | As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the 𝐿2 distance between two 𝐿2-normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the 𝐿2 distance between certain 𝐿1 unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle. |
| format | Article |
| id | doaj-art-7f7a963f0c1e4ac6bb55c0be368fffc9 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-7f7a963f0c1e4ac6bb55c0be368fffc92025-02-03T05:51:41ZengWileyJournal of Applied Mathematics1110-757X1687-00422010-01-01201010.1155/2010/464815464815Some Remarks on Diffusion DistancesMaxim J. Goldberg0Seonja Kim1Theoretical and Applied Science, Ramapo College of NJ, 505 Ramapo Valley Road, Mahwah, NJ 07430, USAMathematics Department, SUNY Rockland Community College, 145 College Road, Suffern, NY 10901, USAAs a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the 𝐿2 distance between two 𝐿2-normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the 𝐿2 distance between certain 𝐿1 unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle.http://dx.doi.org/10.1155/2010/464815 |
| spellingShingle | Maxim J. Goldberg Seonja Kim Some Remarks on Diffusion Distances Journal of Applied Mathematics |
| title | Some Remarks on Diffusion Distances |
| title_full | Some Remarks on Diffusion Distances |
| title_fullStr | Some Remarks on Diffusion Distances |
| title_full_unstemmed | Some Remarks on Diffusion Distances |
| title_short | Some Remarks on Diffusion Distances |
| title_sort | some remarks on diffusion distances |
| url | http://dx.doi.org/10.1155/2010/464815 |
| work_keys_str_mv | AT maximjgoldberg someremarksondiffusiondistances AT seonjakim someremarksondiffusiondistances |