Active Data Mining of Correspondence for Qualitative Assessment of - PowerPoint PPT Presentation

Active Data Mining of Correspondence for Qualitative Assessment of Scientific Computations Chris Bailey-Kellogg Purdue Computer Sciences http://www.cs.purdue.edu/homes/cbk/ Naren Ramakrishnan Virginia Tech Computer Science http://people.cs.vt.edu/~ramakris/

Data-Driven Characterization of Scientific Computations • Choice of solver depends on problem characteristics (e.g. matrix sensitivity) and algorithm performance (e.g. convergence). • Empirical characterization (rather than analytical) appropriate under imperfect domain knowledge, lack of theory. Low-level computational experiments �→ high-level properties. • Example: spectral portrait illustrates eigenstructure under perturbations of different magnitudes. Eigenvalues inside given curve 2.5 5 2 2 5 are indistinguishable under 1.5 4 5 3 6 1 6 7 0.5 perturbation of that magnitude. 7 8 8 9 2 8 10 9 8 6 10 10 0 5 9 7 9 6 4 8 7 −0.5 3 Suggests numerical precision 7 6 −1 6 5 −1.5 5 necessary. 2 −2 5 4 3 −2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Active Data Mining with SAL Abstract Description • Spatial aggregation (bottom-up): uniform operators and data types for extracting multi-layer struc- Higher-Level Objects Redescribe Localize tures in spatial data. Equivalence classes • Ambiguity-directed sampling Classify Ambiguities (top-down): focus data collection Interpolate N-graph Sample Spatial Aggregate on difficult choice points. objects Lower-Level Objects • Underlying domain knowledge: Redescribe Localize continuity, locality. Input Field

Simple example: Input points (values not shown) Aggregate (localize computation) 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −2 −1 0 1 2 3 4 5 6 7 −2 −1 0 1 2 3 4 5 6 7 Classify (group connected points with similar-enough value) 3 2 1 0 −1 −2 −3 −2 −1 0 1 2 3 4 5 6 7

Correspondence Extension to SAL • Key idea: identify mutually-reinforcing relationships among features of spatial objects, in order to combat noise and sparsity. • SAL particularly conducive: aggregation composes hierarchical spatial objects:

• Mechanism: 1. Establish analogy as relation among lower-level constituents of higher-level objects. 3 2 1 Ex: adjacent points of 0 neighboring iso-contours −1 −2 −3 −2 −1 0 1 2 3 4 5 6 7 2. Abstract lower-level analogy into higher-level correspondence. Ex: parameterized curve deformation. • Bridge lower-/higher-level gap: analogy’s meaning derived from higher-level context; abstraction enables computation of global properties (containment, breaks, overall quality). • Directly usable in ambiguity-directed sampling to address difficulties in correspondence.

Application 1: Matrix Spectral Portrait Analysis Spectral portrait for matrix A plots complex map: P ( z ) = log 10 �A� 2 � ( A − zI ) − 1 � 2 2.5 5 2 2 5 1.5 5 4 3 6 6 1 7 0.5 7 8 8 9 10 9 2 6 8 10 8 9 10 0 5 7 9 6 4 8 7 −0.5 3 7 6 −1 6 5 −1.5 5 2 −2 5 4 3 −2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Singularities at eigenvalues; level curves capture “equivalent” eigenvalues wrt perturbations (i.e. curve k contains eigenvalues of all perturbed matrices A + E for E a matrix with � E � 2 ≤ k �A� 2 ). Perturbation-equivalence indicates sensitivity to numerical error. Ex: 2 & 3 most sensitive, then 4, then 1.

Correspondence-Based Merge Identification Approach: compute merge tree, indicating perturbation levels at which eigenvalues become indistinguishable, by finding correspondences among curves. 1. Sample perturbation levels on regular grid; interpolate iso-curves. 3 1 0.8 2 0.6 0.4 1 0.2 0 0 −0.2 −1 −0.4 −0.6 −2 −0.8 −1 −3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2 −1 0 1 2 3 4 5 6 7

2. Aggregate curve points in Delaunay triangulation. 3 1 0.8 2 0.6 0.4 1 0.2 0 0 −0.2 −1 −0.4 −0.6 −2 −0.8 −1 −3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2 −1 0 1 2 3 4 5 6 7 3. Analogy: cross-curve edges in triangulation. 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 4.5

4. Correspondence abstraction: merge events in tree. 1 2 2 3 3 4 4 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 (4,0) (2,0) (3,0) (1,0) (4,0) (3,0) (2,0) (1,0) 5. Evaluate confidence in correspondence: fractions of points matched, angular separation between “separating” samples. 6. Sample to ensure curve locations adequately constrained by separating samples (so couldn’t have merged at smaller perturbation level).

Results • (2 n − 3)!! possible binary merge trees; most not explicitly considered (would be low confidence). • Initial grid: one sample between each eigenvalue, one unit larger than bounding box. • Subsample or expand grid when merge events poorly separated. • Tested on variety of polynomial companion matrices: different numbers / spacings of roots. • High-confidence model selection after 1–3 subsamples, 1–3 grid expansions. • Substantially less computation than “one-size-fits-all”; confidence metric and explainability.

Application 2: Matrix Jordan Form Jordan form analysis: • Input: matrix A of dimension n , r ≤ n independent eigenvectors with eigenvalues λ i of multiplicity ρ i . • Jordan decomposition: r upper triangular “blocks”     J 1 λ i 1     J 2 λ i 1     B − 1 AB = , J i =         · · 1         J r λ i • Typical algorithms numerically unstable.

Graphical Analysis of Jordan Form • Infer multiplicity by eigenvalue perturbations: iφ 1 ρi e λ i + | δ | ρi • Phase φ of perturbation δ ranges over multiples of π ⇒ computed values are vertices of regular 2 ρ i -gon, centered on λ i , with diameter from | δ | . • Ex: 8-by-8 Brunet matrix with structure ( − 1) 1 ( − 2) 1 (7) 3 (7) 3 , focusing on Jordan block for first (7) 3 : −3 x 10 1.5 1 0.5 0 −0.5 −1 −1.5 6.998 6.9985 6.999 6.9995 7 7.0005 7.001 7.0015 7.002

Correspondence-Based Symmetry Analysis Approach: compute Jordan structure by identifying portrait symmetry (i.e. auto-correspondence), abstracting as rotation by π/ρ around eigenvalue. 1. Sample points by random normwise perturbation at magnitude(s) of interest. 0.1 0.1 0 0 −0.1 −0.1 6.9 7 7.1 6.9 7 7.1 2. Aggregate triples into triangles.

3. Analogy among triangle vertices by congruence (computed via geometric hashing). 0.1 0.1 0 0 −0.1 −0.1 6.9 7 7.1 6.9 7 7.1 4. Correspondence as rotation ( x, y, θ ) overlaying vertices of congruent triangles. 0.1 0.1 0 0 −0.1 −0.1 6.9 7 7.1 6.9 7 7.1 Eigenvalue=(7.00,0.00); Eigenvalue=(7.00,0.00); rotation=60.46 ◦ ; ρ =3 rotation=60.13 ◦ ; ρ =3

5. Evaluate confidence in correspondence: distance between points and partners, regularity of sides of polygon. 6. Sample when entropy of models is high.

Results • 10 matrices; 4-10 perturbation levels; 6-8 samples each round. • Vary # models generated by varying congruence tolerance. • Three sample collection policies: 1. Collect at same level: 1.0–2.7 rounds 2. Collect at next higher level: better when #1 uses low level. 3. Collect at same level until begin “hallucinating”: better for Brunet-type matrices. • Symmetry quickly eliminates bad models. • No real advantage to varying perturbation level (independent estimates, irresp. of level). • Small amount of computation required for high-confidence assessment of Jordan form.

Some Related Work • F. Chaitin-Chatelin and V. Frayss´ e: graphical analysis of scientific computations (spectral portraits). • A. Edelman and Y. Ma: Jordan perturbation phenomena. • X. Huang and F. Zhao: correspondence in weather data iso-contours. • Lots of work in vision on computing and tracking correspondence. • D.A. Cohn, Z. Chahramani, and M.I. Jordan: active learning.

Discussion • Correspondence mechanism within Spatial Aggregation leverages hierarchical spatial objects and relationships. • First systematic algorithms for performing complete imagistic analyses (not relying on human visual inspection) of matrix eigenstructure. • Efficient, focused sampling and iterative model evaluation until high confidence obtained. • Overcome noise and sparsity by utilizing locality and continuity to identify mutually-reinforcing interpretations. • Many thanks to reviewers! • Funding: CBK (NSF IIS-0237654) and NR (NSF EIA-9974956, EIA-9984317, and EIA-0103660).