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We systematically clarify the parameter domains that guarantee monotonicity, concavity, and invertibility, derive series and integral representations, and provide explicit links to a broad class of one- and two-parameter deformations, including Tsallis, Kaniadakis, Schw\\\"ammle--Tsallis, Kaniadakis--Scarfone, and Tempesta-type logarithms and their inverse exponentials. In this way, the Euler $(a,b)$-logarithm is established as a unifying kernel for a wide family of generalized entropies and divergence measures.\n  On the algorithmic side, we extend applications of the Euler logarithm to modern machine learning and optimization. We introduce generalized Exponentiated Gradient (GEG) and Mirror Descent (MD) schemes in which the Euler $(a,b)$-logarithm acts","title":"Generalized Euler Logarithm and its Applications in Machine Learning: Natural Gradient, Backpropagation, Generalized EG, Mirror Descent and OLPS","url":"https://arxiv.org/abs/2502.17500","vendor":"arxiv_cs_ai"},"summary":"arXiv:2502.17500v3 Announce Type: replace-cross \nAbstract: This paper investigates in depth the fundamental properties of the two-parameter generalized Euler logarithm and its inverse, the associated deformed $(a,b)$-exponential function. We systematically clarify the parameter domains that guarantee monotonicity, concavity, and invertibility, derive series and integral representations, and provide explicit links to a broad class of one- and two-parameter deformations, including Tsallis, Kaniadakis, Schw\\\"ammle--Tsallis, Kaniadakis--Scarfone, and Tempesta-type logarithms and their inverse exponentials. In this way, the Euler $(a,b)$-logarithm is established as a unifying kernel for a wide family of generalized entropies and divergence measures.\n  On the algorithmic side, we extend applications of the Euler logarithm to modern machine learning and optimization. We introduce generalized Exponentiated Gradient (GEG) and Mirror Descent (MD) schemes in which the Euler $(a,b)$-logarithm acts","title":"Generalized Euler Logarithm and its Applications in Machine Learning: Natural Gradient, Backpropagation, Generalized EG, Mirror Descent and OLPS","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-11T04:43:50Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2502.17500"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:824c4356f901a0d7135374af2247f71835f689404e92bb43ff766b5b0e067e8d0da8a45c90d37004b88b5b8290dcc0763897d1e002652cb170360c82dc421306","signer":"crovia.substrate","subject":{"observed_at":"2026-05-11T04:43:50Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2502.17500"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"8671525c4d11128c12befe1b1410e21a8a79e033441630562d1d3af27426c591","leaf_index":126617,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"89c2aec021fbc20854ad1d07fac51be88c39e07c25e9ae792e785f8ea8cd2110","side":"left"},{"sibling":"3a106be29686c0edbe75d78a5c542fc8ac484f39e2698d53026246edf6a22829","side":"right"},{"sibling":"679b1c314d555871ca063d39bd5e3fee45f800c5c2c29ed82a8889cb0988c711","side":"right"},{"sibling":"4e6afd5b8be79017a798c8a8dbf58f988a547282dfb05f3d5e50a9e7195a937b","side":"left"},{"sibling":"e22c8ba6b8bae5ce514d420bbb035e62d33998b91a58969fc2ac2dea94512b6f","side":"left"},{"sibling":"6c9e8c5a1e345bc6b3af276de43297191dcb4d602c6f58aca9836ebca554c2db","side":"right"},{"sibling":"54aeb1b325b6ed32e0e93fc93a91c18b716eb6c01be8986d8e913f5ba02e2683","side":"right"},{"sibling":"8145236c073312ace1276c28162c95dcc0d2fd49b8c636293fac87efad823f12","side":"left"},{"sibling":"62d48549e4d4466089a57a96bd9bea15741d62a7cb314f0f9c37c4c22ac650fa","side":"right"},{"sibling":"a2c696699a233359c7b4b418ebd356db453d4b886f4bd076ef34b15ee86144a3","side":"left"},{"sibling":"1a581be91236d1f25e8d47fe5efa0e2b51b7f0f6d706ef76a9093067688e5d56","side":"left"},{"sibling":"878cc30108509c9fa1fc52705a216f519d73b647916fcbdfc30a389934d3364b","side":"left"},{"sibling":"ae7dfff36ba07d9f48c36c28a341482ef244ee13cd0efd49aee2bbef2fd65f87","side":"right"},{"sibling":"62ac6554017807bd83187f5a3e5f4f72d6c482616429c2780e9fff1f4845fa04","side":"left"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_da2eccc5c5c5d8885431903360026c050c9b35f0c4d00c2c621ea888f73a8d48"}}