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In the main theorem, irrespective of the size of $\\C$, we give the following estimate for the $k$th largest non-trivial singular value of the normalized table: $s_k \\le 9\\disc_{k } (\\C ) (k+2 -9k\\ln \\disc_{k } (\\C ))$, provided $\\disc_{k } (\\C ) <1$ and $k\\le \\rk (\\C )$. This statement is the converse of Theorem 7 of Bolla \\cite{Bol"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.6443","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-27T15:47:27Z","cross_cats_sorted":[],"title_canon_sha256":"13fc66db4f79bb9f5e1afe6211a975beb3162fdc52a2fb29f74c8156a84188e8","abstract_canon_sha256":"17c973e287b62dad70b7095361f085cd153f10ef44a6b95688fbe406d0d55d27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:11.670991Z","signature_b64":"FffvkrkLQvtD2Ec25DEfZO9wvXIDI6opw8hii1GqehjqRSay/vrzbAdInsUbYFIUaBwHINsa+EbWs7pGP/YkCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1903172bee5c513de466b56442c9bedc8e65f1eee6db0bd1d51f17d90ff4ab7","last_reissued_at":"2026-05-18T02:28:11.670369Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:11.670369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relating multiway discrepancy and singular values of graphs and contingency tables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marianna Bolla","submitted_at":"2014-08-27T15:47:27Z","abstract_excerpt":"The $k$-way discrepancy $\\disc_k (\\C)$ of a rectangular array $\\C$ of nonnegative entries is the minimum of the maxima of the within- and between-cluster discrepancies that can be obtained by simultaneous $k$-clusterings (proper partitions) of its rows and columns. In the main theorem, irrespective of the size of $\\C$, we give the following estimate for the $k$th largest non-trivial singular value of the normalized table: $s_k \\le 9\\disc_{k } (\\C ) (k+2 -9k\\ln \\disc_{k } (\\C ))$, provided $\\disc_{k } (\\C ) <1$ and $k\\le \\rk (\\C )$. 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