Metric embeddings, graph expansion, and high-dimensional convex geometry James R. Lee Institute for Advanced Study.
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metric embeddings, graph expansion,
and high-dimensional convex geometry
James R. LeeInstitute for Advanced Study
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
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
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).
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!
geometric approach
SDP relaxation:
triangle inequality constraints:
A distance function satisfying theabove constraint is called a negative-type metric on V.
geometric approach
SDP relaxation:
triangle inequality constraints:
impose a strange geometry on the solution space
geometric approach
triangle inequality constraints:
impose a strange geometry on the solution space
·
x
yz
1Euclidean distance after t steps
is at most √ t
The distortion of f is the smallest number D such that
embeddings and distortion
Given two metric spaces (X,dX) and (Y,dY), an embedding of X into Y is a mapping f : X ! Y.
The distortion of f is the smallest number D such that
embeddings and distortion
Given two metric spaces (X,dX) and (Y,dY), an embedding of X into Y is a mapping f : X ! Y.
We will be concerned with the cases Y = L1 or Y = L2
(think of Y = Rn with the L1 or L2 norm)
In this case, we write c1(X) or c2(X) for the smallest possible distortion necessary to embed X into L1 and L2, resp.
the connection
negative-type metrics (NEG), embeddings, L1, L2
NEG metric(V,d)
Integrality gap
max c1(V,d)
max distortion into L1
==
allow weights w(u,v)“sparsest cut”
embedding NEG spaces
So we just need to figure out a way to embed every NEG space into L1 with small distortion...
Problem: We don’t have strong L1-specific techniques.
Let’s instead try to embed NEG spaces into L2 spaces (i.e. Euclideanspaces). This is actually stronger, since L2 µ L1, but there is a naturalbarrier... Even the d-dimensional hypercube {0,1}d requires d1/2 = (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 c2(V,d) = O(log n) [Bourgain]
Conjecture: Every n-point NEG metric (V,d) has c2(V,d) = O(√log n)
embedding NEG spaces
Conjecture: Every n-point NEG metric (V,d) has c2(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?
average distortion (1)
The embedding comes in three steps
1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! Rn which sends a “large fraction” of pairs far apart.
NEG space Euclidean space
fRn
average distortion (1)
The embedding comes in three steps
1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! Rn which sends a “large fraction” of pairs far apart.
Every non-expansive map from {0,1}d into L2 mapsmost pairs to distance at most √d = √log n
) average distance contracts by a √log n factor
fRn
hype
rcub
e
average distortion (1)
The embedding comes in three steps
1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! Rn which sends a “large fraction” of pairs far apart.
¼1
|A| ¸ n/5
|B| ¸ n/5
d(A,B) ¸ 1/√log n0
AB
1/√log n
f : X ! R
average distortion (1)
The embedding comes in three steps
1.Average distortion: Fighting concentration of measure Want a non-expansive map f : X ! Rn which sends a “large fraction” of pairs far apart.
¼1
|A| ¸ n/5
|B| ¸ n/5
d(A,B) ¸ 1/√log n
Theorem: Such sets A,B µ X always exist! [Arora-Rao-Vazirani]
single-scale distortion (2)
2. Single-scale distortion Now we want a non-expansive map f : X ! Rn which “handles” all the pairs x,y 2 X with d(x,y)¼1.
¼1
If we had a randomized procedure forgenerating A and B, then we could samplek = O(log n) random coordinates of theform x ! d(x, A), and handle every paira constant fraction of the time(with high probability)...
A
B
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.”
Rn
H want d(A0,B0) ¸ 1/√log n
A0
B0
Choosing A and B “at random”
single-scale distortion (2)
Randomized version:
A
B
2. Prune the “exceptions.”
Rn
H
A0
B0
Pruning ) d(A,B) is large.The hard part is showing that
|A|, |B| = (n) whpafter the pruning!
Choosing A and B “at random”
single-scale distortion (2)
Randomized version:
A
B
Rn
H
Pruning ) d(A,B) is large.The hard part is showing that
|A|, |B| = (n) whpafter the pruning!
[ARV] gives
[L] yields the optimal bound
A and B are not “random” enough
single-scale distortion (2)
Adversarial noise:
A
B
Rn
H
A0, B0 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 points3. 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.
multi-scale distortion (3)
3. Multi-scale distortion Finally, we want to take our analysis of “one scale” and get a low-distortion embedding.
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...
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)!
multi-scale distortion (3)
[Krauthgamer-L-Mendel-Naor, LA, Arora-L-Naor, LB]
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.
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.
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
multi-scale distortion (3)
GLUING THEOREMSIf such an ensemble exists, then X embeds in a Euclidean space with distortion...
[KLMN, LA] (CGR)
[ALN]
[LB]
lower bounds, hardness, and stability
No hardness of approximation results are known for edge expansionunder standard assumptions (e.g. P NP).
Recently, there have been hardness results proved using variantsof Khot’s Unique Games Conjecture (UGC):
[Khot-Vishnoi, Chawla-Krauthgamer-Kumar-Rabani-Sivakumar]
And unconditional results about embeddings, and the integralitygap of the SDP:
[Khot-Vishnoi, Krauthgamer-Rabani]
lower bounds, hardness, and stability
The analysis of all these lower bounds are based on isoperimetric stabilityresults in graphs based on the discrete cube {0,1}d.
Classical fact: The cuts with minimal (S) are dimension cuts.
lower bounds, hardness, and stability
The analysis of all these lower bounds are based on isoperimetric stabilityresults 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)
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