{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:O35CB7KF5MHLVQP6YXTKGQJ4E2","short_pith_number":"pith:O35CB7KF","canonical_record":{"source":{"id":"1307.5870","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-07-22T20:23:29Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"48920d17e0d2f1f41162cec694d88e2c00ebc957492f516c0bdd28d364d134f9","abstract_canon_sha256":"c3900b9dacb3754e00fd4fa56c97ef05fdf2f866885078920ceb0c78fd47cbaf"},"schema_version":"1.0"},"canonical_sha256":"76fa20fd45eb0ebac1fec5e6a3413c26b74d91fac14446d933b746198032b5aa","source":{"kind":"arxiv","id":"1307.5870","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5870","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5870v2","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5870","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"pith_short_12","alias_value":"O35CB7KF5MHL","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"O35CB7KF5MHLVQP6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"O35CB7KF","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:O35CB7KF5MHLVQP6YXTKGQJ4E2","target":"record","payload":{"canonical_record":{"source":{"id":"1307.5870","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-07-22T20:23:29Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"48920d17e0d2f1f41162cec694d88e2c00ebc957492f516c0bdd28d364d134f9","abstract_canon_sha256":"c3900b9dacb3754e00fd4fa56c97ef05fdf2f866885078920ceb0c78fd47cbaf"},"schema_version":"1.0"},"canonical_sha256":"76fa20fd45eb0ebac1fec5e6a3413c26b74d91fac14446d933b746198032b5aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:55.474268Z","signature_b64":"jP7Eo83cb9wc4YdFzpp+xg8F6sXUDZKk/gfDjcTAgVczXoTmZVGqfIIgTsXVX3X/W8w0ClgxJN5OouvKdqv3DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76fa20fd45eb0ebac1fec5e6a3413c26b74d91fac14446d933b746198032b5aa","last_reissued_at":"2026-05-18T03:15:55.473457Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:55.473457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.5870","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":"2026-05-18T03:15:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HO3FnpjC8e7c/F14McUWqjZtgIf8jY4w7teZdxZCABJHDbaU+uSuGtihuxTRtY5/iDnN7J/WlHw+niMnO2U0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:07:16.045917Z"},"content_sha256":"9a3bb4ce87cdbabc5cb6e1c32e787bb592c73d5673b3ecd853e634c1d3b98cd8","schema_version":"1.0","event_id":"sha256:9a3bb4ce87cdbabc5cb6e1c32e787bb592c73d5673b3ecd853e634c1d3b98cd8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:O35CB7KF5MHLVQP6YXTKGQJ4E2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Bo Huang, Cun Mu, Donald Goldfarb, John Wright","submitted_at":"2013-07-22T20:23:29Z","abstract_excerpt":"Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a $K$-way tensor of length $n$ and Tucker rank $r$ from Gaussian measurements requires $\\Omega(r n^{K-1})$ observations. In contrast, a certain (intractable) nonconvex formulation needs only $O(r^K + nrK)$ observations. We introduce a very simple, new convex relaxation, whi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5870","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":"2026-05-18T03:15:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jNiVOVDIw+Qu0pOx+FQCllqbPP7MNKejQiO8akKDsY1B33OmWMYQDdzoRWPx/TncLLCN1nBmgqWv6cmUswX7DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:07:16.046261Z"},"content_sha256":"654c15853561569bb003143cb432a321d06685a8f8589dc19557755a94cf91c3","schema_version":"1.0","event_id":"sha256:654c15853561569bb003143cb432a321d06685a8f8589dc19557755a94cf91c3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/bundle.json","state_url":"https://pith.science/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/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":"2026-05-28T09:07:16Z","links":{"resolver":"https://pith.science/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2","bundle":"https://pith.science/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/bundle.json","state":"https://pith.science/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O35CB7KF5MHLVQP6YXTKGQJ4E2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:O35CB7KF5MHLVQP6YXTKGQJ4E2","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":"c3900b9dacb3754e00fd4fa56c97ef05fdf2f866885078920ceb0c78fd47cbaf","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-07-22T20:23:29Z","title_canon_sha256":"48920d17e0d2f1f41162cec694d88e2c00ebc957492f516c0bdd28d364d134f9"},"schema_version":"1.0","source":{"id":"1307.5870","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5870","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5870v2","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5870","created_at":"2026-05-18T03:15:55Z"},{"alias_kind":"pith_short_12","alias_value":"O35CB7KF5MHL","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"O35CB7KF5MHLVQP6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"O35CB7KF","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:654c15853561569bb003143cb432a321d06685a8f8589dc19557755a94cf91c3","target":"graph","created_at":"2026-05-18T03:15:55Z","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":"Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a $K$-way tensor of length $n$ and Tucker rank $r$ from Gaussian measurements requires $\\Omega(r n^{K-1})$ observations. In contrast, a certain (intractable) nonconvex formulation needs only $O(r^K + nrK)$ observations. We introduce a very simple, new convex relaxation, whi","authors_text":"Bo Huang, Cun Mu, Donald Goldfarb, John Wright","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-07-22T20:23:29Z","title":"Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5870","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":"sha256:9a3bb4ce87cdbabc5cb6e1c32e787bb592c73d5673b3ecd853e634c1d3b98cd8","target":"record","created_at":"2026-05-18T03:15:55Z","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":"c3900b9dacb3754e00fd4fa56c97ef05fdf2f866885078920ceb0c78fd47cbaf","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-07-22T20:23:29Z","title_canon_sha256":"48920d17e0d2f1f41162cec694d88e2c00ebc957492f516c0bdd28d364d134f9"},"schema_version":"1.0","source":{"id":"1307.5870","kind":"arxiv","version":2}},"canonical_sha256":"76fa20fd45eb0ebac1fec5e6a3413c26b74d91fac14446d933b746198032b5aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76fa20fd45eb0ebac1fec5e6a3413c26b74d91fac14446d933b746198032b5aa","first_computed_at":"2026-05-18T03:15:55.473457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:55.473457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jP7Eo83cb9wc4YdFzpp+xg8F6sXUDZKk/gfDjcTAgVczXoTmZVGqfIIgTsXVX3X/W8w0ClgxJN5OouvKdqv3DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:55.474268Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5870","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a3bb4ce87cdbabc5cb6e1c32e787bb592c73d5673b3ecd853e634c1d3b98cd8","sha256:654c15853561569bb003143cb432a321d06685a8f8589dc19557755a94cf91c3"],"state_sha256":"819ceeeff871c7e1f85db60e216c7d55379de5ca962b4c2a9b5bf546ed080459"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d2UU2aQ/crgmBcB2B/W6V8Xj7l5ZHk+hfWMw1ZNmK+0FpQNjk7exl8myjCJ3adp7R28NnkW5tQTFUa94np+fDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T09:07:16.048234Z","bundle_sha256":"7ffa9ba1aaf5a084806601345730184d3ac704cd5991ce3d3d07275b36c79474"}}