Cover’s theorem
E641828
Cover’s theorem is a result in statistical pattern recognition stating that data cast nonlinearly into a higher-dimensional space is more likely to be linearly separable than in a lower-dimensional space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cover’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7115740 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cover’s theorem Context triple: [Thomas M. Cover, theoremNamedAfter, Cover’s theorem]
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A.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
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B.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
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C.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
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D.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
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E.
Jarník–Besicovitch theorem
The Jarník–Besicovitch theorem is a fundamental result in metric number theory that determines the Hausdorff dimension of sets of real numbers that are very well approximable by rationals.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cover’s theorem Target entity description: Cover’s theorem is a result in statistical pattern recognition stating that data cast nonlinearly into a higher-dimensional space is more likely to be linearly separable than in a lower-dimensional space.
-
A.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
B.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
-
C.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
-
D.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
-
E.
Jarník–Besicovitch theorem
The Jarník–Besicovitch theorem is a fundamental result in metric number theory that determines the Hausdorff dimension of sets of real numbers that are very well approximable by rationals.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
result in statistical pattern recognition
ⓘ
theorem ⓘ |
| addresses | conditions for linear separability ⓘ |
| appearsIn |
literature on statistical learning theory
ⓘ
textbooks on pattern recognition ⓘ |
| appliesTo |
classification problems
ⓘ
pattern classification ⓘ |
| assumes | random nonlinear embedding of data into higher-dimensional space ⓘ |
| assumption | patterns are in general position ⓘ |
| concerns |
number of dichotomies realizable by hyperplanes
ⓘ
randomly placed points in general position ⓘ |
| conclusion | patterns are more likely to be linearly separable in higher-dimensional spaces ⓘ |
| context |
binary classification
ⓘ
multiclass classification via one-vs-rest schemes ⓘ |
| coreIdea | nonlinear mapping to higher-dimensional spaces increases probability of linear separability ⓘ |
| describes | relationship between dimensionality and linear separability of patterns ⓘ |
| field |
machine learning
ⓘ
pattern recognition ⓘ statistical pattern recognition ⓘ |
| formalizes | probability of linear separability as a function of dimensionality and number of patterns ⓘ |
| implies | increased dimensionality can simplify classification boundaries ⓘ |
| influenced |
development of kernel trick
ⓘ
development of support vector machines ⓘ theory of pattern classification in high dimensions ⓘ |
| mathematicalForm | bound on number of linearly separable labelings of points ⓘ |
| motivates |
use of high-dimensional embeddings in classification
ⓘ
use of nonlinear feature maps ⓘ |
| namedAfter | Thomas M. Cover NERFINISHED ⓘ |
| originallyPublishedIn | IEEE Transactions on Electronic Computers NERFINISHED ⓘ |
| originalTitle | Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition NERFINISHED ⓘ |
| publicationYear | 1965 ⓘ |
| relatedTo |
curse of dimensionality
ⓘ
feature space transformation ⓘ kernel methods ⓘ linear separability ⓘ support vector machines NERFINISHED ⓘ |
| statedBy | Thomas M. Cover NERFINISHED ⓘ |
| states | for a complex pattern-classification problem, a nonlinear transformation to a high-dimensional space is likely to convert it into a linearly separable problem ⓘ |
| typeOf | geometric result in high-dimensional spaces ⓘ |
| usedIn |
analysis of high-dimensional feature mappings
ⓘ
design of kernel-based classifiers ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cover’s theorem Description of subject: Cover’s theorem is a result in statistical pattern recognition stating that data cast nonlinearly into a higher-dimensional space is more likely to be linearly separable than in a lower-dimensional space.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.