{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VDKB4CABCZR4Q3YC275L62VXPL","short_pith_number":"pith:VDKB4CAB","canonical_record":{"source":{"id":"1512.09302","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-12-31T14:57:03Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"1c441539fd85722a676a5c035effb3b19c84c20aea1755142c62851c03a793a9","abstract_canon_sha256":"eb0c38df3a629eaab020b43555f81ed5bcbfe55fc05ec8e0afc651547dff8f1c"},"schema_version":"1.0"},"canonical_sha256":"a8d41e08011663c86f02d7fabf6ab77af65bab70bbf4c86aeffdf87b9673d385","source":{"kind":"arxiv","id":"1512.09302","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.09302","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1512.09302v2","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.09302","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"VDKB4CABCZR4","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VDKB4CABCZR4Q3YC","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VDKB4CAB","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VDKB4CABCZR4Q3YC275L62VXPL","target":"record","payload":{"canonical_record":{"source":{"id":"1512.09302","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-12-31T14:57:03Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"1c441539fd85722a676a5c035effb3b19c84c20aea1755142c62851c03a793a9","abstract_canon_sha256":"eb0c38df3a629eaab020b43555f81ed5bcbfe55fc05ec8e0afc651547dff8f1c"},"schema_version":"1.0"},"canonical_sha256":"a8d41e08011663c86f02d7fabf6ab77af65bab70bbf4c86aeffdf87b9673d385","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:14.787175Z","signature_b64":"/w4n72xRAtDrvOy+9mdBH/Tk98eyEsHm7WmcDzckjdeL1WMbx41gpcYUCLV7FS8qZ5mt9hdnbiRTg0uqr/UaBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8d41e08011663c86f02d7fabf6ab77af65bab70bbf4c86aeffdf87b9673d385","last_reissued_at":"2026-05-18T01:10:14.786681Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:14.786681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.09302","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PRNZ+rO+Po9Cu5oXPAS3BuLYIVz4LeTCY42WOQkGDU+N+eOUVjza/2fwPnL2cIIZDNUPBaGwkVkvNYnv+YNaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:04:16.525270Z"},"content_sha256":"afebd90dafd69d70d6c90ca65cd73dda77c18a6de0862fbe2d79d056b1e5432a","schema_version":"1.0","event_id":"sha256:afebd90dafd69d70d6c90ca65cd73dda77c18a6de0862fbe2d79d056b1e5432a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VDKB4CABCZR4Q3YC275L62VXPL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear Convergence of Proximal Gradient Algorithm with Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"math.OC","authors_text":"Bo Wen, Ting Kei Pong, Xiaojun Chen","submitted_at":"2015-12-31T14:57:03Z","abstract_excerpt":"In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [19] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges $R$-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also $R$-linearly convergent. In addition, the thre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.09302","kind":"arxiv","version":2},"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":"2026-05-18T01:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nWgPHWXfbPWV8ZYFGVEFP/OGKCtF6Mld2OROTi8oITBTcNGoe2C/L0StLA5gY76zDKrZJi8iZPVoYLrWug75CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T20:04:16.525996Z"},"content_sha256":"c0c07bad60ccc3f1719d6b978887bd2de4ceeb7a7deef2a35ce28c65f56b9916","schema_version":"1.0","event_id":"sha256:c0c07bad60ccc3f1719d6b978887bd2de4ceeb7a7deef2a35ce28c65f56b9916"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VDKB4CABCZR4Q3YC275L62VXPL/bundle.json","state_url":"https://pith.science/pith/VDKB4CABCZR4Q3YC275L62VXPL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VDKB4CABCZR4Q3YC275L62VXPL/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":"2026-05-26T20:04:16Z","links":{"resolver":"https://pith.science/pith/VDKB4CABCZR4Q3YC275L62VXPL","bundle":"https://pith.science/pith/VDKB4CABCZR4Q3YC275L62VXPL/bundle.json","state":"https://pith.science/pith/VDKB4CABCZR4Q3YC275L62VXPL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VDKB4CABCZR4Q3YC275L62VXPL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VDKB4CABCZR4Q3YC275L62VXPL","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":"eb0c38df3a629eaab020b43555f81ed5bcbfe55fc05ec8e0afc651547dff8f1c","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-12-31T14:57:03Z","title_canon_sha256":"1c441539fd85722a676a5c035effb3b19c84c20aea1755142c62851c03a793a9"},"schema_version":"1.0","source":{"id":"1512.09302","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.09302","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1512.09302v2","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.09302","created_at":"2026-05-18T01:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"VDKB4CABCZR4","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VDKB4CABCZR4Q3YC","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VDKB4CAB","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:c0c07bad60ccc3f1719d6b978887bd2de4ceeb7a7deef2a35ce28c65f56b9916","target":"graph","created_at":"2026-05-18T01:10:14Z","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 study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [19] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges $R$-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also $R$-linearly convergent. In addition, the thre","authors_text":"Bo Wen, Ting Kei Pong, Xiaojun Chen","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-12-31T14:57:03Z","title":"Linear Convergence of Proximal Gradient Algorithm with Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.09302","kind":"arxiv","version":2},"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:afebd90dafd69d70d6c90ca65cd73dda77c18a6de0862fbe2d79d056b1e5432a","target":"record","created_at":"2026-05-18T01:10:14Z","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":"eb0c38df3a629eaab020b43555f81ed5bcbfe55fc05ec8e0afc651547dff8f1c","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-12-31T14:57:03Z","title_canon_sha256":"1c441539fd85722a676a5c035effb3b19c84c20aea1755142c62851c03a793a9"},"schema_version":"1.0","source":{"id":"1512.09302","kind":"arxiv","version":2}},"canonical_sha256":"a8d41e08011663c86f02d7fabf6ab77af65bab70bbf4c86aeffdf87b9673d385","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8d41e08011663c86f02d7fabf6ab77af65bab70bbf4c86aeffdf87b9673d385","first_computed_at":"2026-05-18T01:10:14.786681Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:14.786681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/w4n72xRAtDrvOy+9mdBH/Tk98eyEsHm7WmcDzckjdeL1WMbx41gpcYUCLV7FS8qZ5mt9hdnbiRTg0uqr/UaBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:14.787175Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.09302","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afebd90dafd69d70d6c90ca65cd73dda77c18a6de0862fbe2d79d056b1e5432a","sha256:c0c07bad60ccc3f1719d6b978887bd2de4ceeb7a7deef2a35ce28c65f56b9916"],"state_sha256":"e8b8eb7637a8f48ef4fbc17f2336e5c23ec7a20ab12b961fb1fcd8c469a3a0f2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"65Hj4lEf1bdBbGvDn+ckbNL5h2cMNAi9aq2P/VbMQTUD10vfbMQgUCLVjfvgIsiKkZ50zXsYPZzMirvmucKdBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T20:04:16.529846Z","bundle_sha256":"bb1c92544197781f712cd836d2bf9aceac5b2c556fbd6e1ad83075f26fb219ff"}}