Earthmover resilience & testing in ordered structures

Transcription

1 Earthmover resilience & testing in ordered structures Omri Ben-Eliezer Tel-Aviv University Eldar Fischer Technion Computational Complexity Conference 28 UCSD, San Diago

2 Property testing (RS96, GGR98) Meta problem: Given property P, efficiently distinguish between Objects that satisfy P Objects that are far from satisfying P

3 Graph property testing Definition: An!-test for a property P is given query access to an unknown graph G on n vertices, and acts as follows. " satis'ies ) G is!-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

4 Graph property testing Query = Is there an edge between u and v? (dense graph model) " satis'ies ) G is!-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

5 Graph property testing!-far = need to add/remove!" # edges in G to satisfy P. (dense graph model) $ satis)ies + G is!-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

6 Graph property testing Definition: A property P is testable if it has an!-test making "! queries for any! >. Question (GGR98): Which graph properties are testable?

7 Canonical tests An!-test is canonical if it queries a random induced subgraph and accepts/rejects only based on queried subgraph.

8 Canonical tests Theorem [AFKS, GT3]: P testable ó P canonically testable Intuition: Original test makes! queries Canonical test picks random 2! vertices, then simulates original test

9 Tolerant testing [PRR 6] Test is!, # -tolerant ( & < () if it acts as follows. Motivation: Noisy input ) satis.ies G is δ close to P G is #-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

10 Tolerant testing [PRR 6] P is tolerantly testable " $ : P has a $, & -test making '(&) queries. + satisies 2 G is δ close to P G is *-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

11 Distance estimation P is estimable " # : P has a " #, " -test making &(#, ") queries. G is () *) close to P G is )-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

12 Testing vs tolerant testing vs distance estimation Theorem [Fischer, Newman 5]: For graph properties, P canonically testable P estimable " satis'ies ) G is!-far from P G is (! δ) close to P G is!-far from P ACCEPT DON T CARE REJECT ACCEPT DON T CARE REJECT

13 Summary - graph properties testability canonical testability trivial " satis'ies ) G is!-far from P (GT3) " satis'ies ) G is!-far from P ACCEPT DON T CARE REJECT ACCEPT DON T CARE REJECT estimability tolerant testability (FN5) G (! δ) close to P G is!-far from P (FN5) G δ close to P G is!-far from P ACCEPT DON T CARE REJECT ACCEPT DON T CARE REJECT

14 What about ordered structures? Strings (D) Images (2D) AKA ordered matrices Vertex-ordered graphs (2D) and hypergraphs Hypercube (high-d): a different story...

15 Image property testing Unknown!! image # over fixed set of pixels Σ Query = What is the color of pixel in location (i,j)? # satis*ies, # is %-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

16 Image property testing Unknown!! image # over fixed set of pixels Σ %-far = need to modify %& ' pixels in # to satisfy P # satis,ies. # is %-far from P ACCEPT (with prob. 2/3) DON T CARE REJECT (with prob. 2/3)

17 Image property testing canonical test = pick randomly! rows and! columns, query all pixels in intersection. queried pixel

18 String property testing Query access to unknown string of length! over fixed alphabet Σ. canonical test = pick randomly # elements and query them. queried element

19 What about ordered structures? Do similar characterizations hold for ordered structures? No, testability/estimability canonical testability Example: not containing three consecutive -s in / strings. No, testability tolerant testability. [Fischer, Fortnow 5] Properties based on codes & PCPPs. Yes, for global enough properties. [This work]

20 Earthmover resilience (strings) Flip operation: Flip locations of neighboring entries Definition: Earthmover distance between strings S and S is! " #, # % = ' () min{ number of flips to create S from S, } Definition: Property P is earthmover resilient if -:, (,) s.t. String # satisfies P String # A B, B % -(D) String # is D-close to P

21 Earthmover resilience (images) Flip operation: Flip locations of neighboring rows/columns Definition: Earthmover distance between images! and! is # $ %, % = ( )* min{ number of flips to create! from!, } Definition: Property P is earthmover resilient if.:, (,) s.t. image % satisfies P image % satisfies Image % is B-close to P

22 Which properties are earthmover resilient? All unordered graph properties [trivial] All hereditary properties of strings, images & ordered graphs [AKNS, ABF7] Global visual properties of images Convexity of the s s form a half plane [This work]: In general, all properties with sparse boundary between s and s. Convex shape of s Monotonicity: a hereditary property

23 Earthmover resilience vs canonical testing [This work]: For string properties P, P earthmover resilient P canonically testable For image and ordered graph properties P, P earthmover resilient P tolerantly testable P canonically testable

24 Canonical testing to estimation [This work]: For image and ordered graph properties P, P canonically testable P (canonically) estimable Corollary [ABF7 + This work]: P hereditary P (canonically) estimable

25 ER properties are similar to graph properties For earthmover resilient properties of images / ordered graphs: Tolerant testability estimability Canonical testability

26 Warmup proof: ER canonical testing in binary strings ER => piecewise canonical testing Consider Interval partition of string into sufficiently many parts. In each interval, make sufficiently many random queries to estimate number of s and s. Due to ER, this gives good estimate for distance to P:!"#$%&'( ), + min 3 VD(S, S8 ) Where VD(S,S ) denotes average variation distance between the distributions of s and s in each interval.

27 Warmup proof: ER canonical testing in binary strings piecewise canonical testing => canonical testing Interval partition can be approximated by Picking sufficiently many random queries. Partitioning them artificially into intervals. Consequently, piecewise canonical tests can be simulated by canonical ones.

28 Bits from the proof: Szemerédi regularity lemma [Szemerédi 75]: Any graph has an equipartition of size!(#), so that almost all pairs of parts are #-regular. density D Pair is '-regular if ( * # for any pair of subsets of size #& Size #& Size N density d Size #& Size N

29 Bits from the proof: canonical testing --> estimation High level idea - unordered case [Fischer Newman 5] Step : If P is canonically testable, densities of small induced subgraphs among graphs satisfying P different from those of graphs far from P. Step 2: regular partitions of graphs satisfying P differ from graphs far from P.

30 Bits from the proof: canonical testing --> estimation High level idea - unordered case [Fischer Newman 5] Step : If P is canonically testable, densities of small induced subgraphs among graphs satisfying P different from those of graphs far from P. Step 2: regular partitions of graphs satisfying P differ from graphs far from P. Step 3: Estimating which regular partitions a graph has - doable with constant number of queries. Step 4: distance of G from P min distance of a regular partition for G from a regular partition for P.

31 Bits from the proof: canonical testing --> estimation Our observation Above scheme essentially works for multipartite graphs. Given ordered graph!, take interval partition of the vertices, effectively approximating! by a multipartite graph.

32 Bonus: Regular reducibility [Alon, Fischer, Newman, Shapira 6]: A graph property P is canonically testable P can be described using regular partitions [This work]: Same holds for images and ordered graphs.

33 Towards a limit object? Proofs in [ABF 7] and this work rely on interval partitioning. A limit object (graphon-like [BCLSSV5; LS8; BCLSV8]) for images and ordered graphs via interval partitioning?

34 Other open questions Testability + earthmover resilience canonical testability? More efficient conversions from testability to estimability Hereditary properties in graphs [Hoppen, Kohayakawa, Lang, Lefmann, Stagni 6 + 7] The landscape of property testing Global properties seem easy to test [AFKS, FN, ABF 7, this work] Local properties are easy to test [BKR 7, B 8+] Algebraic structure makes it hard to test [FF 5, FPS 7] Other general results?