{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:RILSMFYPD5QRICKPKXHG24HUW6","short_pith_number":"pith:RILSMFYP","canonical_record":{"source":{"id":"1709.10441","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-29T15:01:43Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"959b2dfb8b8d381d11deb03a3fe73c72f40d4bef4365832b76bc31ba3f2d58cb","abstract_canon_sha256":"300ce4baa82bc89c1b37a531a8488f50e1dcfcc8199071bb048855b7bc52ba7a"},"schema_version":"1.0"},"canonical_sha256":"8a1726170f1f6114094f55ce6d70f4b78a52611b3dc4bd7ebb6a59bdc407cfdd","source":{"kind":"arxiv","id":"1709.10441","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.10441","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"arxiv_version","alias_value":"1709.10441v3","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.10441","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"pith_short_12","alias_value":"RILSMFYPD5QR","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RILSMFYPD5QRICKP","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RILSMFYP","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:RILSMFYPD5QRICKPKXHG24HUW6","target":"record","payload":{"canonical_record":{"source":{"id":"1709.10441","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-29T15:01:43Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"959b2dfb8b8d381d11deb03a3fe73c72f40d4bef4365832b76bc31ba3f2d58cb","abstract_canon_sha256":"300ce4baa82bc89c1b37a531a8488f50e1dcfcc8199071bb048855b7bc52ba7a"},"schema_version":"1.0"},"canonical_sha256":"8a1726170f1f6114094f55ce6d70f4b78a52611b3dc4bd7ebb6a59bdc407cfdd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:00.847330Z","signature_b64":"8dy/w55fWDA7JqwWUNIMrelAHNcvFYKhjFUuVc14yKboe7WANYePav0LqDM+XBwqtjpVYCi+ANmbv7g+8jlgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a1726170f1f6114094f55ce6d70f4b78a52611b3dc4bd7ebb6a59bdc407cfdd","last_reissued_at":"2026-05-18T00:14:00.846468Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:00.846468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.10441","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hju9ZckW46iUjJW7QgIlXBXMAm9FrXHxLGMgQrxkTt1gkPSad+Hk1My/+DFUOe/bGNIoBhouOqkEWf9uCZ2SDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:18:16.287791Z"},"content_sha256":"1cd1b2808a54ca05cdbb00c92b51ede6708b9de5f84c90e05ea86a45ebc15f85","schema_version":"1.0","event_id":"sha256:1cd1b2808a54ca05cdbb00c92b51ede6708b9de5f84c90e05ea86a45ebc15f85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:RILSMFYPD5QRICKPKXHG24HUW6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A representer theorem for deep kernel learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"cs.LG","authors_text":"Bastian Bohn, Christian Rieger, Michael Griebel","submitted_at":"2017-09-29T15:01:43Z","abstract_excerpt":"In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical foundation for the analysis of machine learning algorithms based on compositions of functions. As a direct consequence in the finite-sample case, the corresponding infinite-dimensional minimization problems can be recast into (nonlinear) finite-dimensional minimization problems, which can be tackled with nonlinear optimization algorithms. Moreover, we show how concatenated"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.10441","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PKUoSYwlmjKx+EtHQxMLFS1Uqv/QBmoRaLcKe5LGyOd2Xvrpu4Wl9IxydJrut/JnUgfCLFSa2dh/RL+lvfU5Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:18:16.288380Z"},"content_sha256":"d093313ba9782e51d85d84266c10969e45612e5295fee0ac5dff6e269603510f","schema_version":"1.0","event_id":"sha256:d093313ba9782e51d85d84266c10969e45612e5295fee0ac5dff6e269603510f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RILSMFYPD5QRICKPKXHG24HUW6/bundle.json","state_url":"https://pith.science/pith/RILSMFYPD5QRICKPKXHG24HUW6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RILSMFYPD5QRICKPKXHG24HUW6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T23:18:16Z","links":{"resolver":"https://pith.science/pith/RILSMFYPD5QRICKPKXHG24HUW6","bundle":"https://pith.science/pith/RILSMFYPD5QRICKPKXHG24HUW6/bundle.json","state":"https://pith.science/pith/RILSMFYPD5QRICKPKXHG24HUW6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RILSMFYPD5QRICKPKXHG24HUW6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:RILSMFYPD5QRICKPKXHG24HUW6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"300ce4baa82bc89c1b37a531a8488f50e1dcfcc8199071bb048855b7bc52ba7a","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-29T15:01:43Z","title_canon_sha256":"959b2dfb8b8d381d11deb03a3fe73c72f40d4bef4365832b76bc31ba3f2d58cb"},"schema_version":"1.0","source":{"id":"1709.10441","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.10441","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"arxiv_version","alias_value":"1709.10441v3","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.10441","created_at":"2026-05-18T00:14:00Z"},{"alias_kind":"pith_short_12","alias_value":"RILSMFYPD5QR","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RILSMFYPD5QRICKP","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RILSMFYP","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:d093313ba9782e51d85d84266c10969e45612e5295fee0ac5dff6e269603510f","target":"graph","created_at":"2026-05-18T00:14:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical foundation for the analysis of machine learning algorithms based on compositions of functions. As a direct consequence in the finite-sample case, the corresponding infinite-dimensional minimization problems can be recast into (nonlinear) finite-dimensional minimization problems, which can be tackled with nonlinear optimization algorithms. Moreover, we show how concatenated","authors_text":"Bastian Bohn, Christian Rieger, Michael Griebel","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-29T15:01:43Z","title":"A representer theorem for deep kernel learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.10441","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1cd1b2808a54ca05cdbb00c92b51ede6708b9de5f84c90e05ea86a45ebc15f85","target":"record","created_at":"2026-05-18T00:14:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"300ce4baa82bc89c1b37a531a8488f50e1dcfcc8199071bb048855b7bc52ba7a","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-29T15:01:43Z","title_canon_sha256":"959b2dfb8b8d381d11deb03a3fe73c72f40d4bef4365832b76bc31ba3f2d58cb"},"schema_version":"1.0","source":{"id":"1709.10441","kind":"arxiv","version":3}},"canonical_sha256":"8a1726170f1f6114094f55ce6d70f4b78a52611b3dc4bd7ebb6a59bdc407cfdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8a1726170f1f6114094f55ce6d70f4b78a52611b3dc4bd7ebb6a59bdc407cfdd","first_computed_at":"2026-05-18T00:14:00.846468Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:00.846468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8dy/w55fWDA7JqwWUNIMrelAHNcvFYKhjFUuVc14yKboe7WANYePav0LqDM+XBwqtjpVYCi+ANmbv7g+8jlgDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:00.847330Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.10441","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cd1b2808a54ca05cdbb00c92b51ede6708b9de5f84c90e05ea86a45ebc15f85","sha256:d093313ba9782e51d85d84266c10969e45612e5295fee0ac5dff6e269603510f"],"state_sha256":"ed4ad911d7752545085b44e452613ce10fe107bbedb8c5b62c5bf7c1431ef889"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7q4mfJJGVw1HRxfEsDmbIZyWsUoAkl5+fXZB4GnHt5dnKGWNFhQS8DmAAIBLVHoTxa9EVuap8LaiLGhImBcUDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:18:16.291421Z","bundle_sha256":"c1864b6b763afdbf9840a9f71acfe0bf5f210615fda2f8c72998dc7c6cbaaacb"}}