Motivation Minkowski-type Metrics for Sets Distances Between Sets Based on Set Commonality Kathy Horadam and Michael Nyblom RMIT University, Melbourne, Australia Australian and New Zealand Mathematics Convention, Melbourne 10 December, 2014 Horadam
Motivation Minkowski-type Metrics for Sets Outline Motivation 1 Biometric Graph Matching Scoring functions Minkowski-type Metrics for Sets 2 Minkowski-type metrics for sets Normalising the Minkowski-type metrics for sets Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Outline Motivation 1 Biometric Graph Matching Scoring functions Minkowski-type Metrics for Sets 2 Minkowski-type metrics for sets Normalising the Minkowski-type metrics for sets Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Background A novel area for me, probably a once-off! Work arises from a problem in biometric matching Biometric matching: Personal identification through “what you are" (fingerprint, face) not “what you know" (PIN, password) or “what you carry" (token, smartcard) Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Biometric matching Biometric samples from an individual vary each time they are presented Need error-tolerant or fuzzy matching of biometric features to authenticate But not too error-tolerant or fuzzy, or an imposter can be authenticated as you! Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Example: Features of Retina biometric Extracting graphs from biometric images: A retina image (a) and its retina graph (b) (a) (b) 0 −100 −200 −300 −400 −500 −600 200 400 600 Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Comparing biometric graphs Use error-correcting graph matching algorithm to compare retina graphs G and G ∗ Optimal edit path defines Maximum Common Subgraph (mcs) G ∩ G ∗ Can count many types of structural elements of each graph G , G ∗ , G ∩ G ∗ eg for vertices: we count | V | , | V ∗ | and | V ∩ V ∗ | . Many other subgraphs could be counted: # edges, # paths length 2, # nodes degree 3, # simple cycles, etc etc Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Outline Motivation 1 Biometric Graph Matching Scoring functions Minkowski-type Metrics for Sets 2 Minkowski-type metrics for sets Normalising the Minkowski-type metrics for sets Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Scoring genuine samples against imposters 10 8 6 MATCH density Genuine Imposter 4 2 0 0.2 0.4 0.6 0.8 1.0 DSQRT � Example: use d sqrt ( G , G ∗ ) = | V ∩ V ∗ | / | V | | V ∗ | as the scoring function. d sqrt is a distance but NOT a metric (preferred) Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions What metrics are known for comparing sets? Distance or dissimilarity d : X × X → R on data set X non-negative, symmetric and reflexive function. Normalised if d ( u , v ) ≤ 1 ∀ u , v ∈ X . And is a metric if ∀ u , v , w ∈ X , d ( u , v ) = 0 ⇔ u = v ; and the triangle inequality holds: d ( u , v ) ≤ d ( u , w ) + d ( w , v ) . Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions What metrics are known for comparing sets? Notation. U � = ∅ , X = set of finite nonempty subsets of U . For X i , X j , X k ∈ X , let x i = | X i | , x ij = | X i ∩ X j | , x ijk = | X i ∩ X j ∩ X k | . Let m ij = x i − x ij , so x i > 0 and m ij ≥ 0. Put x i = x ∗ i + y ij + y ik + x ijk , where y ij = y ji = x ij − x ijk , so m ij = x i − x ij = x ∗ i + y ik . Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions What metrics are known for comparing sets? The following are known normalised metrics on X . The Jaccard, or set-difference, metric d sd ( X i , X j ) = ( m ij + m ji ) / ( x i + x j − x ij ) = 1 − x ij / ( x i + x j − x ij ) . The normalised maximum metric d max ( X i , X j ) = max { m ij , m ji } / max { x i , x j } = 1 − x ij / max { x i , x j } . CAN WE FIND MORE SET METRICS BASED ON SET COMMONALITY ? Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Which measures separate genuine from imposter retinas best? Score Distance? Metric? NN √ d sqrt (good-ish) × 0 . 0486 √ √ d sd (good-ish) 0 . 0154 √ √ d max (ugly) − 0 . 0450 Comparison of nearest neighbour (NN) distances of vertex sets in VARIA retina database for 3 scoring functions ARE THERE BETTER SCORING METRICS THAN d max ? Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Which measures separate genuine from imposter retinas best? Score Distance? Metric? NN √ d sqrt (good-ish) × 0 . 0486 √ √ d sd (good-ish) 0 . 0154 √ √ d max (ugly) − 0 . 0450 Comparison of nearest neighbour (NN) distances of vertex sets in VARIA retina database for 3 scoring functions ARE THERE BETTER SCORING METRICS THAN d max ? Horadam
Motivation Biometric Graph Matching Minkowski-type Metrics for Sets Scoring functions Which measures separate genuine from imposter retinas best? Score Distance? Metric? NN √ d sqrt (good-ish) × 0 . 0486 √ √ d sd (good-ish) 0 . 0154 √ √ d max (ugly) − 0 . 0450 Comparison of nearest neighbour (NN) distances of vertex sets in VARIA retina database for 3 scoring functions ARE THERE BETTER SCORING METRICS THAN d max ? Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets Outline Motivation 1 Biometric Graph Matching Scoring functions Minkowski-type Metrics for Sets 2 Minkowski-type metrics for sets Normalising the Minkowski-type metrics for sets Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets The Minkowski or p -norm metrics d p on R n These are defined for real p ≥ 1 and dimension n ≥ 1 to be n � 1 | u i − v i | p ) p . d p (( u 1 , u 2 , . . . , u n ) , ( v 1 , v 2 , . . . , v n )) = ( i = 1 p = 1: absolute value distance, (taxicab, city-block or Manhattan distance). p = 2: usual Euclidean distance. lim p →∞ d p = d ∞ : infinity norm (Chebyshev) distance, max distance d ∞ (( u 1 , u 2 , . . . , u n ) , ( v 1 , v 2 , . . . , v n )) = max {| u 1 − v 1 | , | u 2 − v 2 | , . . . , | u n − v n |} . Varying p changes the weight given to larger and smaller differences. Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets Eureka! Minkowski-type Metric Family for Sets The following definition gives set-based metrics with analogous properties. For each p ≥ 1, define d 2 , p : X × X → R to be 1 d 2 , p ( X i , X j ) = [ m p ij + m p p . ji ] Theorem d 2 , p is a metric ; 1 d 2 , 1 = d sd , lim p →∞ d 2 , p = d max and if p < p ′ , d 2 , p ≥ d 2 , p ′ . 2 For sets, the Minkowski-type metric is a modification of the 2D real Minkowski metric. d 2 , p ( X i , X j ) = d p (( x i , x j ) , ( x ij , x ij )) . d 2 , 2 is analogous to the Euclidean metric d 2 in the plane. Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets Outline Motivation 1 Biometric Graph Matching Scoring functions Minkowski-type Metrics for Sets 2 Minkowski-type metrics for sets Normalising the Minkowski-type metrics for sets Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets Normalising the Minkowski-type Metric Family Theorem For each p ≥ 1 , define d 2 , p to be d 2 , p ( X i , X j ) = [ m p ij + m p ji ] 1 / p / ( x ij + [ m p ij + m p ji ] 1 / p ) . Then d 2 , p is a normalised metric on X ; 1 d 2 , 1 = d sd , (Jaccard metric) 2 lim p →∞ d 2 , p = d max (maximum metric) 3 d 2 , p ( X i , X j ) = 1 ⇔ X i ∩ X j = ∅ ; 4 d 2 , p is monotone decreasing in p. 5 Horadam
Motivation Minkowski-type metrics for sets Minkowski-type Metrics for Sets Normalising the Minkowski-type metrics for sets Normalising the Minkowski-type Metric Family Lemma Reduction Lemma. The triangle inequality holds for X i , X j , X k if and only if it holds for the subsets X ′ i ⊆ X i , X ′ j ⊆ X j and X ′ k ⊆ X k where X ′ i = [ X i \ ( X i ∩ X j )] ∪ ( X i ∩ X j ∩ X k ) , X ′ j = [ X j \ ( X i ∩ X j )] ∪ ( X i ∩ X j ∩ X k ) similarly, and X ′ k = ( X i ∩ X k ) ∪ ( X j ∩ X k ) . (only if) d 2 , p ( X i , X j ) ≤ d 2 , p ( X ′ i , X ′ j ) since ji ) 1 / p ] − 1 ≤ [ 1 + x ijk / ( m p [ 1 + x ij / ( m p ij + m p ij + m p ji ) 1 / p ] − 1 , d 2 , p ( X ′ i , X ′ k ) + d 2 , p ( X ′ j , X ′ k ) ≤ d 2 , p ( X i , X k ) + d 2 , p ( X j , X k ) since i ) p + y p jk ) 1 / p ] − 1 ≤ [ 1 + x ik / ( m p ki ) 1 / p ] − 1 ; ik + m p [ 1 + x ik / (( x ∗ and by symmetry j ) p + y p ik ) 1 / p ] − 1 ≤ [ 1 + x jk / ( m p jk + m p kj ) 1 / p ] − 1 . [ 1 + x jk / (( x ∗ � Horadam
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