1. metric embeddings, graph expansion, and high-dimensional convex geometry James R. Lee Institute for Advanced Study

2. graph expansion and the sparsest cut Given a graph G =( V,E), and a subset S µ V, we denote S E(S, S) The edge expansion of G is the value

3. graph expansion and the sparsest cut Given a graph G =( V,E), and a subset S µ V, we denote S E(S, S) The edge expansion of G is the value

4. graph expansion and the sparsest cut Given a graph G =( V,E), and a subset S µ V, we denote S E(S, S) The edge expansion of G is the value Goal: Find the least-expanding cut in G (at least approximately).

5. geometric approach There is a natural SDP-based approach: Spectral analysis (first try) can be computed by a semi-definite program gap can be  (n) even if G is an n-cycle!

6. geometric approach SDP relaxation:

7. geometric approach SDP relaxation: triangle inequality constraints: A distance function satisfying the above constraint is called a negative-type metric on V.

8. geometric approach SDP relaxation: triangle inequality constraints: impose a strange geometry on the solution space

9. geometric approach triangle inequality constraints: impose a strange geometry on the solution space ·   x y z 1 Euclidean distance after t steps is at most √ t

10. The distortion of f is the smallest number D such that embeddings and distortion Given two metric spaces (X,d X ) and (Y,d Y ), an embedding of X into Y is a mapping f : X ! Y.

11. The distortion of f is the smallest number D such that embeddings and distortion Given two metric spaces (X,d X ) and (Y,d Y ), an embedding of X into Y is a mapping f : X ! Y. We will be concerned with the cases Y = L 1 or Y = L 2 (think of Y = R n with the L 1 or L 2 norm) In this case, we write c 1 (X) or c 2 (X) for the smallest possible distortion necessary to embed X into L 1 and L 2, resp.

12. the connection negative-type metrics (NEG), embeddings, L 1, L 2 NEG metric (V,d) Integrality gap max c 1 (V,d) max distortion into L 1 = = allow weights w(u,v) “sparsest cut”

13. embedding NEG spaces So we just need to figure out a way to embed every NEG space into L 1 with small distortion... Problem: We don’t have strong L 1 -specific techniques. Let’s instead try to embed NEG spaces into L 2 spaces (i.e. Euclidean spaces). This is actually stronger, since L 2 µ L 1, but there is a natural barrier... Even the d-dimensional hypercube {0,1} d requires d 1/2 = (log n) 1/2 distortion to embed into a Euclidean space. GOAL: Prove that the hypercube is the “worst” NEG metric. Known: Every n-point NEG metric (V,d) has c 2 (V,d) = O(log n) [Bourgain] Conjecture: Every n-point NEG metric (V,d) has c 2 (V,d) = O(√log n)

14. embedding NEG spaces Conjecture: Every n-point NEG metric (V,d) has c 2 (V,d) = O(√log n) Implies O(√log n)-approximation for edge expansion (even SparsestCut), improving the previous O(log n) bound. [Leighton-Rao, Linial-London-Rabinovich, Aumann-Rabani] Also: Something provable to be gained from spectral approach! Thinking about the conjecture: Subsets of hypercubes {0,1} k provide interesting NEG metrics. If you had to pick an n-point subset of some hypercube which is furthest from a Euclidean space, would you just choose {0,1} log n, or a sparse subset of some higher-dimensional cube?

15. average distortion (1) The embedding comes in three steps 1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! R n which sends a “large fraction” of pairs far apart. NEG space Euclidean space f RnRn

16. average distortion (1) The embedding comes in three steps 1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! R n which sends a “large fraction” of pairs far apart. Every non-expansive map from {0,1} d into L 2 maps most pairs to distance at most √ d = √ log n ) average distance contracts by a √ log n factor f RnRn hypercube

17. average distortion (1) The embedding comes in three steps 1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! R n which sends a “large fraction” of pairs far apart. ¼1¼1 | A | ¸ n/5 | B | ¸ n/5 d(A,B) ¸ 1 /√log n 0 A B 1 /√log n f : X ! R

18. average distortion (1) The embedding comes in three steps 1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! R n which sends a “large fraction” of pairs far apart. ¼1¼1 | A | ¸ n/5 | B | ¸ n/5 d(A,B) ¸ 1 /√log n Theorem: Such sets A,B µ X always exist! [Arora-Rao-Vazirani]

19. single-scale distortion (2) 2. Single-scale distortion Now we want a non-expansive map f : X ! R n which “handles” all the pairs x,y 2 X with d(x,y) ¼ 1. ¼1¼1 If we had a randomized procedure for generating A and B, then we could sample k = O(log n) random coordinates of the form x ! d(x, A), and handle every pair a constant fraction of the time (with high probability)... A B

20. Choosing A and B “at random” single-scale distortion (2) Randomized version: A B 1.Choose a random (n- 1 )-hyperplane. 2. Prune the “exceptions.” RnRn H want d(A 0,B 0 ) ¸ 1 /√log n A0A0 B0B0

21. Choosing A and B “at random” single-scale distortion (2) Randomized version: A B 2. Prune the “exceptions.” RnRn H A0A0 B0B0 Pruning ) d(A,B) is large. The hard part is showing that | A |, | B | =  (n) whp after the pruning!

22. Choosing A and B “at random” single-scale distortion (2) Randomized version: A B RnRn H Pruning ) d(A,B) is large. The hard part is showing that | A |, | B | =  (n) whp after the pruning! [ARV] gives [L] yields the optimal bound

23. A and B are not “random” enough single-scale distortion (2) Adversarial noise: A B RnRn H A 0, B 0 would be great, but we are stuck with A,B [Chawla-Gupta-Racke] (multiplicative update): 1.Give every point of X some weight. 2. Make it harder to prune heavy points 3. If a point is not pruned in some iteration, half its weight. 4. The adversary cannot keep pruning the same point from the matching. After O(log n) iterations, every point is left un-pruned in at least ½ of the trials.

24. multi-scale distortion (3) 3. Multi-scale distortion Finally, we want to take our analysis of “one scale” and get a low-distortion embedding.

25. multi-scale distortion (3) metric spaces have various scales

26. multi-scale distortion (3) 3. Multi-scale distortion Need an embedding that handles all scales simultaneously. So far, we know that if (X,d) is an n-point NEG metric, then...

27. multi-scale distortion (3) 3. Multi-scale distortion Known: Using some tricks, the number of “relevant” scales is only m = O(log n), so take the corresponding maps.... and just “concatenate” the coordinates and rescale: Oops: The distortion of this map is only O(log n)!

28. multi-scale distortion (3) [Krauthgamer-L-Mendel-Naor, L A, Arora-L-Naor, L B ] Basic moral: Not all scales are created equal. (measured descent, gluing lemmas, etc.) x The local expansion of a metric space plays a central role. Ratio small ) locality well-behaved. Represents the “dimension” of X near x 2 X at scale R. Key fact: X has only n points.

29. multi-scale distortion (3) x The local expansion of a metric space plays a central role. Ratio small ) locality well-behaved. Represents the “dimension” of X near x 2 X at scale R. Key fact: X has only n points.

30. multi-scale distortion (3) The local expansion of a metric space plays a central role. Ratio small ) locality well-behaved. Represents the “dimension” of X near x 2 X at scale R. Key fact: X has only n points. controls smoothness of bump functions (useful for gluing maps on a metric space) controls size of “accurate” random samples

31. multi-scale distortion (3) G LUING T HEOREMS If such an ensemble exists, then X embeds in a Euclidean space with distortion... [KLMN, L A ] (CGR) [ALN] [L B ]

32. lower bounds, hardness, and stability No hardness of approximation results are known for edge expansion under standard assumptions (e.g. P  NP). Recently, there have been hardness results proved using variants of Khot’s Unique Games Conjecture (UGC): [Khot-Vishnoi, Chawla-Krauthgamer-Kumar-Rabani-Sivakumar] And unconditional results about embeddings, and the integrality gap of the SDP: [Khot-Vishnoi, Krauthgamer-Rabani]

33. lower bounds, hardness, and stability The analysis of all these lower bounds are based on isoperimetric stability results in graphs based on the discrete cube {0,1} d. Classical fact: The cuts with minimal  (S) are dimension cuts.

34. lower bounds, hardness, and stability The analysis of all these lower bounds are based on isoperimetric stability results in graphs based on the discrete cube {0,1} d. Stability version: Every near-optimal cut is “close” to a dimension cut. (much harder: uses discrete Fourier analysis)

35. open problems What is the right bound for embedding NEG metrics into L 1 ? Does every planar graph metric embed into L 1 with O( 1 ) distortion? (Strongly related to “multi-scale gluing” for L 1 embeddings) What about embedding edit distance into L 1 ? (Applications to sketching, near-neighbor search, etc.)

36. graph expansion and the sparsest cut The S PARSEST C UT problem This is a “non-uniform” notion of expansion. The uniform case w(u,v) = 1, u,v 2 V corresponds (essentially) to edge expansion. If we also have a demand function w : V £ V ! R +, one defines where  S is the indicator function of the subset S µ V. The goal is to find the subset S µ V which minimizes  w (S).

37. graph expansion and the sparsest cut The S PARSEST C UT problem If we also have a demand function w : V £ V ! R +, one defines where  S is the indicator function of the subset S µ V. Since this problem is NP-hard, we try instead to find a cut which is close to optimal.