An Analysis of Transformed Unadjusted Langevin Algorithm for Heavy-Tailed Sampling
We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diff...
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| Veröffentlicht in: | IEEE transactions on information theory 2024-01, Vol.70 (1), p.571-593 |
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| container_title | IEEE transactions on information theory |
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| creator | He, Ye Balasubramanian, Krishnakumar Erdogdu, Murat A. |
| description | We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincaré inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations. |
| doi_str_mv | 10.1109/TIT.2023.3318152 |
| format | Article |
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We highlight the relationship between our assumptions and functional inequalities (super and weak Poincaré inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.</description><subject>Algorithms</subject><subject>Complexity of sampling</subject><subject>Complexity theory</subject><subject>Differential equations</subject><subject>Dirichlet problem</subject><subject>functional inequalities</subject><subject>Heavily-tailed distribution</subject><subject>heavy-tailed densities</subject><subject>Indium tin oxide</subject><subject>Inequalities</subject><subject>Isomorphism</subject><subject>Large scale integration</subject><subject>Lightly-tailed distribution</subject><subject>Polynomials</subject><subject>Samplers</subject><subject>Sampling</subject><subject>Symmetric matrices</subject><subject>Tail</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpNkE1LAzEQhoMoWKt3Dx4WPG-dycducixFbaEg6PYc0k22btmPmmwL_fdNaQ-e5h143oF5CHlGmCCCeisWxYQCZRPGUKKgN2SEQuSpygS_JSMAlKniXN6ThxC2ceUC6Yh8T7tk2pnmGOqQ9FVSeNOFqvets8mqM3a7D0OMS9Nt3KGObLPpfT38tkmEkrkzh2NamLqJzI9pd03dbR7JXWWa4J6uc0xWH-_FbJ4uvz4Xs-kyLSkXQyozmTmo7JohL0UFmOeQWymzLEdrkKFjAFm1NpK53CmLpbWCApfGSVUyYGPyerm78_3f3oVBb_u9j78ETRVkQiqV80jBhSp9H4J3ld75ujX-qBH02ZyO5vTZnL6ai5WXS6V2zv3DqZBMITsBPttpQw</recordid><startdate>202401</startdate><enddate>202401</enddate><creator>He, Ye</creator><creator>Balasubramanian, Krishnakumar</creator><creator>Erdogdu, Murat A.</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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| subjects | Algorithms Complexity of sampling Complexity theory Differential equations Dirichlet problem functional inequalities Heavily-tailed distribution heavy-tailed densities Indium tin oxide Inequalities Isomorphism Large scale integration Lightly-tailed distribution Polynomials Samplers Sampling Symmetric matrices Tail |
| title | An Analysis of Transformed Unadjusted Langevin Algorithm for Heavy-Tailed Sampling |
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