{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:KOV7D7MQSGRYLQ2BB24QTT2IYY","short_pith_number":"pith:KOV7D7MQ","canonical_record":{"source":{"id":"1709.05789","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-18T07:05:12Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"ee2331dee008970910341941193998d26d0eb1e20a15cc88e6e1ce7971b0be01","abstract_canon_sha256":"c62e30eda64d1b18a9990581eacb83528ecfae1ee27384451aaad9ffcea979f1"},"schema_version":"1.0"},"canonical_sha256":"53abf1fd9091a385c3410eb909cf48c6327fd4aebd28e638c8643b9687c49f1a","source":{"kind":"arxiv","id":"1709.05789","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05789","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05789v3","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05789","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"pith_short_12","alias_value":"KOV7D7MQSGRY","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"KOV7D7MQSGRYLQ2B","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"KOV7D7MQ","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:KOV7D7MQSGRYLQ2BB24QTT2IYY","target":"record","payload":{"canonical_record":{"source":{"id":"1709.05789","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-18T07:05:12Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"ee2331dee008970910341941193998d26d0eb1e20a15cc88e6e1ce7971b0be01","abstract_canon_sha256":"c62e30eda64d1b18a9990581eacb83528ecfae1ee27384451aaad9ffcea979f1"},"schema_version":"1.0"},"canonical_sha256":"53abf1fd9091a385c3410eb909cf48c6327fd4aebd28e638c8643b9687c49f1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:04.008822Z","signature_b64":"KO5aJse0ggMipqIQAzZ8oGWvwBUB5e+I9EjLl4mIsPcav6WtVoIjI+qu7H9CXU9V5Q7uuXrZHzU/0RAVFKfLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53abf1fd9091a385c3410eb909cf48c6327fd4aebd28e638c8643b9687c49f1a","last_reissued_at":"2026-05-18T00:10:04.008178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:04.008178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.05789","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":"2026-05-18T00:10:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ivn5rLbW3o9lJzbSYHDNmeKB9dA/jIiAU2I8odIqj2dGY+3sOV0YWrKNvkfx5gN/fMFV4RSpSkGf1h9MMS1jBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:57:07.520732Z"},"content_sha256":"44d4eb63de51abb09bc5ab52cb252ef2db5c303af576b81c5a36852b33430b80","schema_version":"1.0","event_id":"sha256:44d4eb63de51abb09bc5ab52cb252ef2db5c303af576b81c5a36852b33430b80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:KOV7D7MQSGRYLQ2BB24QTT2IYY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Restricted Isometry of the Columnwise Khatri-Rao Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chandra R Murthy, Saurabh Khanna","submitted_at":"2017-09-18T07:05:12Z","abstract_excerpt":"The columnwise Khatri-Rao product of two matrices is an important matrix type, reprising its role as a structured sensing matrix in many fundamental linear inverse problems. Robust signal recovery in such inverse problems is often contingent on proving the restricted isometry property (RIP) of a certain system matrix expressible as a Khatri-Rao product of two matrices. In this work, we analyze the RIP of a generic columnwise Khatri-Rao product matrix by deriving two upper bounds for its $k^{\\text{th}}$ order Restricted Isometry Constant ($k$-RIC) for different values of $k$. The first RIC boun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05789","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":"2026-05-18T00:10:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XoLrK9fsWM4lILHUQ8N0CBoTh504k9o47RcVQm83kNRzdsjpOILo+nNw9c8z7EphKNLkEdfbdB4sOWPCESbNCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:57:07.521074Z"},"content_sha256":"d9f0c539bfe842a2e1e4e51a8239ed08248f8a6879062d087f7c2b6dbc10e8d9","schema_version":"1.0","event_id":"sha256:d9f0c539bfe842a2e1e4e51a8239ed08248f8a6879062d087f7c2b6dbc10e8d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/bundle.json","state_url":"https://pith.science/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/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":"2026-06-26T15:57:07Z","links":{"resolver":"https://pith.science/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY","bundle":"https://pith.science/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/bundle.json","state":"https://pith.science/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KOV7D7MQSGRYLQ2BB24QTT2IYY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:KOV7D7MQSGRYLQ2BB24QTT2IYY","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":"c62e30eda64d1b18a9990581eacb83528ecfae1ee27384451aaad9ffcea979f1","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-18T07:05:12Z","title_canon_sha256":"ee2331dee008970910341941193998d26d0eb1e20a15cc88e6e1ce7971b0be01"},"schema_version":"1.0","source":{"id":"1709.05789","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05789","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05789v3","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05789","created_at":"2026-05-18T00:10:04Z"},{"alias_kind":"pith_short_12","alias_value":"KOV7D7MQSGRY","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"KOV7D7MQSGRYLQ2B","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"KOV7D7MQ","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:d9f0c539bfe842a2e1e4e51a8239ed08248f8a6879062d087f7c2b6dbc10e8d9","target":"graph","created_at":"2026-05-18T00:10:04Z","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":"The columnwise Khatri-Rao product of two matrices is an important matrix type, reprising its role as a structured sensing matrix in many fundamental linear inverse problems. Robust signal recovery in such inverse problems is often contingent on proving the restricted isometry property (RIP) of a certain system matrix expressible as a Khatri-Rao product of two matrices. In this work, we analyze the RIP of a generic columnwise Khatri-Rao product matrix by deriving two upper bounds for its $k^{\\text{th}}$ order Restricted Isometry Constant ($k$-RIC) for different values of $k$. The first RIC boun","authors_text":"Chandra R Murthy, Saurabh Khanna","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-18T07:05:12Z","title":"On the Restricted Isometry of the Columnwise Khatri-Rao Product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05789","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":"sha256:44d4eb63de51abb09bc5ab52cb252ef2db5c303af576b81c5a36852b33430b80","target":"record","created_at":"2026-05-18T00:10:04Z","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":"c62e30eda64d1b18a9990581eacb83528ecfae1ee27384451aaad9ffcea979f1","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-18T07:05:12Z","title_canon_sha256":"ee2331dee008970910341941193998d26d0eb1e20a15cc88e6e1ce7971b0be01"},"schema_version":"1.0","source":{"id":"1709.05789","kind":"arxiv","version":3}},"canonical_sha256":"53abf1fd9091a385c3410eb909cf48c6327fd4aebd28e638c8643b9687c49f1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53abf1fd9091a385c3410eb909cf48c6327fd4aebd28e638c8643b9687c49f1a","first_computed_at":"2026-05-18T00:10:04.008178Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:04.008178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KO5aJse0ggMipqIQAzZ8oGWvwBUB5e+I9EjLl4mIsPcav6WtVoIjI+qu7H9CXU9V5Q7uuXrZHzU/0RAVFKfLAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:04.008822Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.05789","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44d4eb63de51abb09bc5ab52cb252ef2db5c303af576b81c5a36852b33430b80","sha256:d9f0c539bfe842a2e1e4e51a8239ed08248f8a6879062d087f7c2b6dbc10e8d9"],"state_sha256":"c972a79d301905c7b8be9168568f72259b98897d020293281f9928af126f71c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V57NVSg6ZKcAwBBUal6hca8I8LDZKKtjHoJo8dw/lv2BxBxZeXe13USj/hnFMhf9DuP78VN52sbpPTvpAdidBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T15:57:07.523030Z","bundle_sha256":"644db86b645763c7b213edfb2d66be9ed858fab71051d4da7da92a817f9b65e3"}}