Unique Completability of Partially Filled Low Rank Positive Semidefinite Matrices Tibor Jord´ an (E¨ otv¨ os University, Budapest) joint work with Bill Jackson (QM U. London) and Shin-ichi Tanigawa (CWI, Amsterdam and RIMS, Kyoto) Graph Theory 2015, Nyborg, Denmark Tibor Jord´ an Unique completability
Frameworks with semisimple underlying graphs A graph G is called semisimple if it contains no parallel edges. (but may contain loops). A d -dimensional framework is a pair ( G , p ), where G = ( V , E ) is a semisimple graph and p : V → R d is a map. Tibor Jord´ an Unique completability
Frameworks with semisimple underlying graphs A graph G is called semisimple if it contains no parallel edges. (but may contain loops). A d -dimensional framework is a pair ( G , p ), where G = ( V , E ) is a semisimple graph and p : V → R d is a map. Tibor Jord´ an Unique completability
Global completability We say that two d -dimensional frameworks ( G , q ) and ( G , p ) are equivalent if � p i , p j � = � q i , q j � (for all ij ∈ E ( G )) (1) and that they are congruent if (1) holds for every pair i , j in V ( G ) (including i , j with i = j ). This is equivalent to saying that q i = Ap i for all i ∈ V for some fixed d × d orthogonal matrix A . We say that a d -dimensional framework ( G , p ) is globally (uniquely) completable in R d if for every d -dimensional framework ( G , q ) which is equivalent to ( G , p ) we have that ( G , q ) and ( G , p ) are congruent. Tibor Jord´ an Unique completability
Global completability We say that two d -dimensional frameworks ( G , q ) and ( G , p ) are equivalent if � p i , p j � = � q i , q j � (for all ij ∈ E ( G )) (1) and that they are congruent if (1) holds for every pair i , j in V ( G ) (including i , j with i = j ). This is equivalent to saying that q i = Ap i for all i ∈ V for some fixed d × d orthogonal matrix A . We say that a d -dimensional framework ( G , p ) is globally (uniquely) completable in R d if for every d -dimensional framework ( G , q ) which is equivalent to ( G , p ) we have that ( G , q ) and ( G , p ) are congruent. Tibor Jord´ an Unique completability
Partially filled positive semidefinite matrices Let M be an n × n positive semidefinite matrix of rank d . Then M = P ⊤ P for some d × n matrix P (that is, M is a Gram matrix). Hence it can be defined by specifying n points in R d (corresponding to the columns p i , 1 ≤ i ≤ n of P ). Thus the entry M [ i , j ] is equal to the scalar product � p i , p j � . Let M be a partially filled n × n positive semidefinite matrix of rank d . The given entries define an undirected graph G = ( V , E ) on V = { 1 , 2 , ... n } in which two vertices i , j are adjacent if and only if M [ i , j ] is given. The graph G and the columns of P give rise to a d -dimensional framework. Tibor Jord´ an Unique completability
Partially filled positive semidefinite matrices Let M be an n × n positive semidefinite matrix of rank d . Then M = P ⊤ P for some d × n matrix P (that is, M is a Gram matrix). Hence it can be defined by specifying n points in R d (corresponding to the columns p i , 1 ≤ i ≤ n of P ). Thus the entry M [ i , j ] is equal to the scalar product � p i , p j � . Let M be a partially filled n × n positive semidefinite matrix of rank d . The given entries define an undirected graph G = ( V , E ) on V = { 1 , 2 , ... n } in which two vertices i , j are adjacent if and only if M [ i , j ] is given. The graph G and the columns of P give rise to a d -dimensional framework. Tibor Jord´ an Unique completability
The PSD matrix completion problem Given a partially filled n × n matrix M and an integer d , is it completable to a PSD matrix of rank d ? How can we determine a completion? is the completion unique? does unique completability depend only on the underlying semisimple graph, provided the set of entries of the d × n matrix P is generic? can we develop combinatorial algorithms (resp. sufficient conditions) for testing (resp. implying) unique completability? Tibor Jord´ an Unique completability
Some combinatorial sufficient conditions A preview of the combinatorial/graph theoretic results (given a partially filled n × n matrix M and an integer d ): Suppose that at least ⌈ n / 2 ⌉ + 3 (resp. ⌈ ( n + d ) / 2 ⌉ + 2) entries are known in each row/column of the matrix. Then the rank-2 (resp. rank- d ) PSD completion of M is unique. Suppose that there are at most n − d − 1 unknown entries. Then the rank- d PSD completion of M is unique. Tibor Jord´ an Unique completability
Some combinatorial sufficient conditions A preview of the combinatorial/graph theoretic results (given a partially filled n × n matrix M and an integer d ): Suppose that at least ⌈ n / 2 ⌉ + 3 (resp. ⌈ ( n + d ) / 2 ⌉ + 2) entries are known in each row/column of the matrix. Then the rank-2 (resp. rank- d ) PSD completion of M is unique. Suppose that there are at most n − d − 1 unknown entries. Then the rank- d PSD completion of M is unique. Tibor Jord´ an Unique completability
Previous work on the PSD matrix completion problem Testing completability is NP-hard in R d for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and M. Cucuringu, 2010). Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control. Tibor Jord´ an Unique completability
Previous work on the PSD matrix completion problem Testing completability is NP-hard in R d for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and M. Cucuringu, 2010). Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control. Tibor Jord´ an Unique completability
Previous work on the PSD matrix completion problem Testing completability is NP-hard in R d for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and M. Cucuringu, 2010). Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control. Tibor Jord´ an Unique completability
Local completability We say that ( G , p ) is locally completable in R d if there exists an open neighborhood N ( p ) of p in R d | V | such that for any q ∈ N ( p ) the equivalence of ( G , q ) to ( G , p ) implies that the two frameworks are congruent (or equivalently, if every continuous motion of the vertices of ( G , p ) in R d which preserves equivalence must also preserve congruence). Tibor Jord´ an Unique completability
Examples in R 1 Tibor Jord´ an Unique completability
Examples in R 1 Tibor Jord´ an Unique completability
Examples in R 1 Tibor Jord´ an Unique completability
Examples in R 1 Tibor Jord´ an Unique completability
Infinitesimal c-motions p : V → R d is called an infinitesimal c-motion of ( G , p ) if A map ˙ � p i , ˙ p j � + � p j , ˙ p i � = 0 ( ij ∈ E ) (2) The | E | × d | V | -matrix representing this system of linear equations with variables ˙ p is the completability matrix of ( G , p ), denoted by C ( G , p ). For example, if G is a graph with V ( G ) = { 1 , 2 , 3 , 4 } and E ( G ) = { 11 , 12 , 23 , 24 } , the completability matrix is: 1 2 3 4 11 2 p 1 0 0 0 12 p 2 p 1 0 0 . 23 0 p 3 p 2 0 24 0 p 4 0 p 2 Tibor Jord´ an Unique completability
Infinitesimal c-motions p : V → R d is called an infinitesimal c-motion of ( G , p ) if A map ˙ � p i , ˙ p j � + � p j , ˙ p i � = 0 ( ij ∈ E ) (2) The | E | × d | V | -matrix representing this system of linear equations with variables ˙ p is the completability matrix of ( G , p ), denoted by C ( G , p ). For example, if G is a graph with V ( G ) = { 1 , 2 , 3 , 4 } and E ( G ) = { 11 , 12 , 23 , 24 } , the completability matrix is: 1 2 3 4 11 2 p 1 0 0 0 12 p 2 p 1 0 0 . 23 0 p 3 p 2 0 24 0 p 4 0 p 2 Tibor Jord´ an Unique completability
Trivial c-motions p : V → R d For any d × d skew-symmetric matrix S , the map ˙ defined by ˙ p i = Sp i for i ∈ V is an infinitesimal c -motion. The infinitesimal c-motions of this kind are called trivial . Therefore, if | V | ≥ d , then � d � rankC ( G , p ) ≤ dn − . (3) 2 The rank of C ( G , p ) is also bounded above by the number of edges in the complete semisimple graph on n vertices. Tibor Jord´ an Unique completability
Infinitesimal completability A framework ( G , p ) is said to be infinitesimally completable if � d � n +1 � � rankC ( G , p ) = dn − when n ≥ d or rankC ( G , p ) = 2 2 when n ≤ d . It is c-independent if rankC ( G , p ) = | E | . Tibor Jord´ an Unique completability
Bipartite frameworks If G is bipartite with vertex bipartition { V 1 , V 2 } , then the map ˙ p p ( j ) = − A T p ( j ) for j ∈ V 2 defined by ˙ p ( i ) = Ap ( i ) for i ∈ V 1 and ˙ is also an infinitesimal c-motion of ( G , p ) for any d × d matrix A . Therefore, rank C ( G , p ) ≤ d | V | − d 2 (4) if G is bipartite with | V i | ≥ d , i = 1 , 2. Tibor Jord´ an Unique completability
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