Towards a calculus for non-linear spectral gaps Manor Mendel, Open University Jointly with Assaf Naor, NYU
Dec ’08
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
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Outline
1
Expanders
2
Background & Motivation
3
RVW Expanders & Zig-Zag Product
4
Discussion
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
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Outline
1
Expanders
2
Background & Motivation
3
RVW Expanders & Zig-Zag Product
4
Discussion
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
3 / 38
Graphs (n, d) graph G = (V , E) is on n vertices. the degree is d. undirected, connected, multigraph (self loops + parallel edges)
(5,4) graph
The normalized adjacency matrix A Aij = #((i, j) ∈ E)/d, A is symmetric doubly stochastic. Its eigenvalues are 1 = λ1 > λ2 ≥ · · · λn ≥ −1. Also define λ = max{λ2 , −λn } = maxi≥2 |λi |. M. Mendel (OUI)
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Spectral Gap Definition (Spectral gap) g(G) = 1 − λ2 (G) Intuition: g(G) > 0 ⇐⇒ G is connected.
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Spectral Gap Definition (Spectral gap) g(G) = 1 − λ2 (G) Intuition: g(G) > 0 ⇐⇒ G is connected.
Definition (Expanders) An infinite family of graphs with uniform positive lower bound on the spectral gap. In this definition complete graphs are expanders. We will usually be interested in constant degree expanders. Their existence is by no means trivial. However
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Spectral Gap Definition (Spectral gap) g(G) = 1 − λ2 (G) Intuition: g(G) > 0 ⇐⇒ G is connected.
Definition (Expanders) An infinite family of graphs with uniform positive lower bound on the spectral gap. In this definition complete graphs are expanders. We will usually be interested in constant degree expanders. Their existence is by no means trivial. However
Theorem (Pinsker, Margulis) Constant degree expanders do exist. M. Mendel (OUI)
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Expanders are Common
Derandomization Networking Coding Metric Geometry Geometric Group Theory
M. Mendel (OUI)
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Dec ’08
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Poincaré Inequality (PI) for Expanders Proposition Given G = (V , E), ∀f : V → R, E |f (x) − f (y )|2 ≤ (1 − λ2 )−1
x,y ∈V
E
(x,y )∈E
|f (x) − f (y )|2 .
Proof (I). By translation, we assume wlog that
P
x
f (x) = 0, so
E |f (x) − f (y )|2 = 2 E f (x)2 .
x,y ∈V
x∈V
On the other hand, 1 X 2X 2 X Axy (f (x) − f (y ))2 = f (x)2 − axy f (x)f (y ). n n n (x,y )∈V
M. Mendel (OUI)
x∈V
Non-linear spectral gaps
x,y ∈V
Dec ’08
7 / 38
Poincaré Inequality (PI) for Expanders Proposition Given G = (V , E), ∀f : V → R, E |f (x) − f (y )|2 ≤ (1 − λ2 )−1
x,y ∈V
E
(x,y )∈E
|f (x) − f (y )|2 .
Proof (II). Using the spectral decomposition f = α1 = hf , v1 i = 0), we have X
axy f (x)f (y ) =
x,y ∈V
=
X x,y ∈V
X
αi αj
i,j≥2
M. Mendel (OUI)
X x∈V
P
i
αi vi , (remembering that
X X axy ( αi vi (x))( αj vj (y )) i
vi (x)λj vj (x) =
j
X
αi2 λi
i≥2
Non-linear spectral gaps
X x∈V
vi (x)2 ≤ λ2
X
f (x)2 .
x∈V
Dec ’08
7 / 38
Redefining g(G): PI Const
The previous analysis is tight: Check by taking f = v2 . We can redefine g(G)−1 =
M. Mendel (OUI)
Ex,y ∈V |f (x) − f (y )|2 . 2 const6=f :V →R E(x,y )∈E |f (x) − f (y )| sup
Non-linear spectral gaps
Dec ’08
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Redefining g(G): PI Const
The previous analysis is tight: Check by taking f = v2 . We can redefine g(G)−1 =
Ex,y ∈V |f (x) − f (y )|2 . 2 const6=f :V →R E(x,y )∈E |f (x) − f (y )| sup
Furthermore g(G)−1 =
sup const6=f :V →L2
M. Mendel (OUI)
Ex,y ∈V kf (x) − f (y )k22 . E(x,y )∈E kf (x) − f (y )k22
Non-linear spectral gaps
Dec ’08
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Outline
1
Expanders
2
Background & Motivation
3
RVW Expanders & Zig-Zag Product
4
Discussion
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
9 / 38
BiLipschitz & Coarse embedding
Let f : (X , ρ) → (Y , ν), satisfying
ν(f (x), f (y )) ≥ ρ(x, y ), ∀x, y ∈ X ,
ν(f (x0 ), f (y0 )) = ρ(x0 , y0 ) for some x0 , y0 ∈ X .
Definition (Distortion) dist(f ) = supx6=y
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ν(f (x),f (y )) ρ(x,y )
cY (X ) = inf{dist(f ) : f : X → Y }.
Non-linear spectral gaps
Dec ’08
10 / 38
BiLipschitz & Coarse embedding
Let f : (X , ρ) → (Y , ν), satisfying
ν(f (x), f (y )) ≥ ρ(x, y ), ∀x, y ∈ X ,
ν(f (x0 ), f (y0 )) = ρ(x0 , y0 ) for some x0 , y0 ∈ X .
Definition (Distortion) dist(f ) = supx6=y
ν(f (x),f (y )) ρ(x,y )
cY (X ) = inf{dist(f ) : f : X → Y }.
Remark More generally, dist(f ) = kf kLip · kf −1 kLip .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
10 / 38
Non embedability of Expanders Claim In (n, d) graphs, most pair of points are at distance Ω(logd n) (counting argument). Therefore, Ex,y ∈V dG (x, y )2 = Ω(log2d n). 2 E(x,y )∈E dG (x, y )
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
11 / 38
Non embedability of Expanders Claim In (n, d) graphs, most pair of points are at distance Ω(logd n) (counting argument). Therefore, Ex,y ∈V dG (x, y )2 = Ω(log2d n). 2 E(x,y )∈E dG (x, y )
Theorem (Linial-London-Rabinovich, Matoušek) If (Gn )n is a family of (n, d) graphs with uniform bound g −1 on the PI constant. Then √ cL2 (Gn ) = Ω( g logd n). (Gn ) is not coarsely embeddable in L2 .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
11 / 38
Non embedability of Expanders Claim In (n, d) graphs, most pair of points are at distance Ω(logd n) (counting argument). Therefore, Ex,y ∈V dG (x, y )2 = Ω(log2d n). 2 E(x,y )∈E dG (x, y )
Theorem (Linial-London-Rabinovich, Matoušek) If (Gn )n is a family of (n, d) graphs with uniform bound g −1 on the PI constant. Then √ cL2 (Gn ) = Ω( g logd n). (Gn ) is not coarsely embeddable in L2 . Non-embedding in other spaces? M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
11 / 38
Non embedability of Expanders Claim In (n, d) graphs, most pair of points are at distance Ω(logd n) (counting argument). Therefore, Ex,y ∈V dG (x, y )2 = Ω(log2d n). 2 E(x,y )∈E dG (x, y )
Theorem (Linial-London-Rabinovich, Matoušek) If (Gn )n is a family of (n, d) graphs with uniform bound g −1 on the PI constant. Then √ cL2 (Gn ) = Ω( g logd n). (Gn ) is not coarsely embeddable in L2 . Non-embedding in other spaces? Generalize PI to those spaces! M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
11 / 38
PI Constant in Abstract Spaces ρ : X × X → [0, ∞) is called symmetric kernel if ρ(x, y ) = ρ(y , x).
Definition (PI Constant) g(G, ρ)−1 =
sup const6=f :V →X
M. Mendel (OUI)
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Non-linear spectral gaps
Dec ’08
12 / 38
PI Constant in Abstract Spaces ρ : X × X → [0, ∞) is called symmetric kernel if ρ(x, y ) = ρ(y , x).
Definition (PI Constant) g(G, ρ)−1 =
sup const6=f :V →X
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Example When (X , ρ) = (R, | · |2 ) or (X , ρ) = (L2 , k · k22 ), g(G, ρ) is the spectral gap. When (X , ρ) = ({0, 1}) or (X , ρ) = (L1 , k · k1 ), g(G, ρ) is (closely related to) the edge expansion.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
12 / 38
PI Constant in Abstract Spaces ρ : X × X → [0, ∞) is called symmetric kernel if ρ(x, y ) = ρ(y , x).
Definition (PI Constant) g(G, ρ)−1 =
sup const6=f :V →X
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Example When (X , ρ) = (R, | · |2 ) or (X , ρ) = (L2 , k · k22 ), g(G, ρ) is the spectral gap. When (X , ρ) = ({0, 1}) or (X , ρ) = (L1 , k · k1 ), g(G, ρ) is (closely related to) the edge expansion. We will care mostly when ρ is a power of a metric. But, there are non trivial facts even in this abstract setting. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
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Abstract PI: Simple Observations Observation (Cheeger-Tanner-Alon-Milman) For any graph G and non trivial (X , ρ), q g(G)−1 ≤ g(G, {0, 1})−1 ≤ g(G, ρ)−1 .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
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Abstract PI: Simple Observations Observation (Cheeger-Tanner-Alon-Milman) For any graph G and non trivial (X , ρ), q g(G)−1 ≤ g(G, {0, 1})−1 ≤ g(G, ρ)−1 .
Observation When (X , ρ) is a metric space, and p ≥ 1, g(G, ρp )−1 ≤ diam(G)p
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
13 / 38
Abstract PI: Simple Observations Observation (Cheeger-Tanner-Alon-Milman) For any graph G and non trivial (X , ρ), q g(G)−1 ≤ g(G, {0, 1})−1 ≤ g(G, ρ)−1 .
Observation When (X , ρ) is a metric space, and p ≥ 1, g(G, ρp )−1 ≤ diam(G)p
Observation When (X , ρ) contains all metric spaces and G is an (n, d) graph, g(G, ρp )−1 ≥ (logd−1 (n/2))p . M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
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Inembeddability in other spaces Proposition If (X , ρ) a metric and p ≥ 1, for which there is an (n, d) graphs (Gn ), such that g(G, ρp )−1 ≤ g −1 , then cX (Gn ) = Ω(log n)
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
14 / 38
Inembeddability in other spaces Proposition If (X , ρ) a metric and p ≥ 1, for which there is an (n, d) graphs (Gn ), such that g(G, ρp )−1 ≤ g −1 , then cX (Gn ) = Ω(log n) Theorems of this flavor are of considerable interest for metric geometry, Banach spaces, and geometric group theory. Spaces which were studied in this context before: Lp spaces [Matoušek] Banach Lattices with cotype [Ozawa][Pisier] Uniform convex Banach spaces [Kasparov, Yu] CAT(0) spaces [Gromov] B-convex Banach spaces [Lafforgue]
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Dec ’08
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Related Previous Results Theorem (Matoušek) When 1 ≤ p ≤ q, for (n, d) graph G, g(G, (`p , k · kpp ))−1 ≤ g(G, (`q , k · kqq ))−1 ≤
4q p g(G, (`p , k
· kpp ))−1/p
q
.
In particular, any expander family is also expander with respect to k · kpp .
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Non-linear spectral gaps
Dec ’08
15 / 38
Related Previous Results Theorem (Matoušek) When 1 ≤ p ≤ q, for (n, d) graph G, g(G, (`p , k · kpp ))−1 ≤ g(G, (`q , k · kqq ))−1 ≤
4q p g(G, (`p , k
· kpp ))−1/p
q
.
In particular, any expander family is also expander with respect to k · kpp .
Theorem (Ozawa, Naor-Rabani) For any Banach lattice (X , k · k), with cotype < ∞, ∃α : [0, ∞) → [0, ∞) s.t. g(G, k · k)−1 ≤ α(g(G)−1 )
M. Mendel (OUI)
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Dec ’08
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B-Convex Spaces
Theorem (Lafforgue) If X is B convex Banach space then there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , k · k2X )−1 ≤ g −1 . The proof is algebraic using strong form of Property (T).
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Non-linear spectral gaps
Dec ’08
16 / 38
Natural questions Question (1.) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ? I.e., when does “expander is expander" ?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
17 / 38
Natural questions Question (1.) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ? I.e., when does “expander is expander" ?
Question (2.) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ? I.e., when do expanders exist?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
17 / 38
Natural questions Question (1.) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ? I.e., when does “expander is expander" ?
Question (2.) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ? I.e., when do expanders exist? Those questions are quite general. We don’t have satisfactory answers.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
17 / 38
Natural questions Question (1.) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ? I.e., when does “expander is expander" ?
Question (2.) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ? I.e., when do expanders exist? Those questions are quite general. We don’t have satisfactory answers. We concentrate on the second question — building a “toolbox" for non-linear spectral gaps. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
17 / 38
Natural questions Question (1.) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ? I.e., when does “expander is expander" ?
Question (2.) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ? I.e., when do expanders exist? Those questions are quite general. We don’t have satisfactory answers. We concentrate on the second question — building a “toolbox" for non-linear spectral gaps. Specifically, tools for constructing RVW expanders. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
17 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube".
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces).
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces). Improving Lafforgue’s expanders to have degree 3.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces). Improving Lafforgue’s expanders to have degree 3. Advancing a framework of non-linear spectral gaps.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces). Improving Lafforgue’s expanders to have degree 3. Advancing a framework of non-linear spectral gaps. Raises interesting questions.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces). Improving Lafforgue’s expanders to have degree 3. Advancing a framework of non-linear spectral gaps. Raises interesting questions. Ball’s extension theorem for Lipschitz function into CAT(0) spaces, via barycentric spaces — unifying uniform convexity and CAT(0).
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Contributions A clean, “abstract" and general analysis of the zigzag product. Geometric condition for the decay of the PI const of powers of graph. A high degree but non-trivial expander graph for B-convex spaces in the form of a quotient of a “noisy Hamming cube". RVW expanders for uniformly convex spaces (weaker than Lafforgue’s expanders for B-convex spaces). Improving Lafforgue’s expanders to have degree 3. Advancing a framework of non-linear spectral gaps. Raises interesting questions. Ball’s extension theorem for Lipschitz function into CAT(0) spaces, via barycentric spaces — unifying uniform convexity and CAT(0). Today: Only the zigzag result. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
18 / 38
Outline
1
Expanders
2
Background & Motivation
3
RVW Expanders & Zig-Zag Product
4
Discussion
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
19 / 38
Review of RVW Expanders I
Reingold, Vadhan , & Wigderson [RVW] introduced the zigzag product, and used it to construct expanders. RVW expanders were the first explicit expanders using “purely" combinatorial methods. Since then, they were used in many applications, some of them impossible using previous methods. Expansion beyond the spectral gap. [CRVW] Symmetric log space = log space [Reingold]. Expander codes
The current work is yet another application of this paradigm.
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Dec ’08
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Review of RVW Expanders II Ingredients: Base graph: A (D 2t , D) graph H, with g(H)−1 ≤
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Non-linear spectral gaps
√
t.
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Review of RVW Expanders II Ingredients: Base graph: A (D 2t , D) graph H, with g(H)−1 ≤
√
t.
Gs
Power operation: Given (n, d) graph is an (n, d s ), with 1 − g(Gs ) = (1 − g(G))s , and hence for odd s, g(Gs )−1 ≤ 2 · max{1, g(G)−1 /s} Power: Improves the PI const g −1 , while increasing the degree.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
21 / 38
Review of RVW Expanders II Ingredients: Base graph: A (D 2t , D) graph H, with g(H)−1 ≤
√
t.
Gs
Power operation: Given (n, d) graph is an (n, d s ), with 1 − g(Gs ) = (1 − g(G))s , and hence for odd s, g(Gs )−1 ≤ 2 · max{1, g(G)−1 /s} Power: Improves the PI const g −1 , while increasing the degree. Zig-zag product: If G is (n1 , d1 )-graph, and H is (n2 , d2 ) graph, and n2 = d1 , then G z H is (n1 n2 , d22 ) graph with g(G z H)−1 ≤ g(G)−1 · g(H)−2 Zig-zag: Increase the size, decrease the degree, controlled increase of g −1 . M. Mendel (OUI)
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Review of RVW Expanders III
The expander family (Gi )i is constructed as follows:
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Review of RVW Expanders III
The expander family (Gi )i is constructed as follows: G0 = H 2
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Review of RVW Expanders III
The expander family (Gi )i is constructed as follows: G0 = H 2 Gi+1 = Git zH
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Review of RVW Expanders III
The expander family (Gi )i is constructed as follows: G0 = H 2 Gi+1 = Git zH
By induction: Gi is (D 2ti , D 2 ) graph with g(Gi+1 )−1 ≤ 2 max{1, g(Gi )−1 /t} · g(H)−2 .
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Non-linear spectral gaps
Dec ’08
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Review of RVW Expanders III
The expander family (Gi )i is constructed as follows: G0 = H 2 Gi+1 = Git zH
By induction: Gi is (D 2ti , D 2 ) graph with g(Gi+1 )−1 ≤ 2 max{1, g(Gi )−1 /t} · g(H)−2 . p Hence, if g(H)−1 ≤ t/2, we have ∀i, g(Gi ) ≤ O(1)
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Zig-Zag Product [RVW]
Given:
(n1 , d1 ) graph G
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(n2 , d2 ) graph H
Non-linear spectral gaps
Dec ’08
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Zig-Zag Product [RVW]
Given:
(n1 , d1 ) graph G
(n2 , d2 ) graph H d1 = n2
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Zig-Zag Product [RVW]
Given:
(n1 , d1 ) graph G
(n2 , d2 ) graph H d1 = n2
Define: K = G z H as follows:
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Zig-Zag Product [RVW]
VK = VG × V H
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Zig-Zag Product [RVW]
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Zig-Zag Product [RVW]
M. Mendel (OUI)
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Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
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Zig-Zag Product [RVW]
M. Mendel (OUI)
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Zig-Zag Product [RVW]
M. Mendel (OUI)
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Zig-Zag Product [RVW]
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Zig-Zag Product [RVW]
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Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
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Zig-Zag Product [RVW]
M. Mendel (OUI)
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Zig-Zag Product [RVW]
M. Mendel (OUI)
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Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
24 / 38
Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
24 / 38
Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
24 / 38
Zig-Zag Product [RVW]
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
24 / 38
Zig-Zag Product [RVW]
G z H is (n1 d1 , d22 ) = (n1 n2 , d22 ) graph. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
24 / 38
PI Const of the Zig-Zag Product
[RVW,RTV] showed
M. Mendel (OUI)
that g(G z H)−1 ≤ g(G)−1 · g(H)−2 .
Non-linear spectral gaps
Dec ’08
25 / 38
PI Const of the Zig-Zag Product
[RVW,RTV] showed
that g(G z H)−1 ≤ g(G)−1 · g(H)−2 .
Theorem ([MN] ) For every symmetric kernel ρ, g(G z H, ρ)−1 ≤ g(G, ρ)−1 · g(H, ρ)−2 .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
25 / 38
PI Const of the Zig-Zag Product
[RVW,RTV] showed
that g(G z H)−1 ≤ g(G)−1 · g(H)−2 .
Theorem ([MN] ) For every symmetric kernel ρ, g(G z H, ρ)−1 ≤ g(G, ρ)−1 · g(H, ρ)−2 . Of interest even if one only cares about “classical" spectral gap.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
25 / 38
PI Const of the Zig-Zag Product
z H)−1 ≤ g(G)−1 · g(H)−2 . [RVW,RTV] showed(*) that g(G
Theorem ([MN] *) For every symmetric kernel ρ, g(G z H, ρ)−1 ≤ g(G, ρ)−1 · g(H, ρ)−2 . Of interest even if one only cares about “classical" spectral gap.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
25 / 38
Double Cover (DC) PI Constant Definition (PI Constant) g(G, ρ)−1 = sup f :V →X f 6=const
M. Mendel (OUI)
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Non-linear spectral gaps
Eρ(x, y) ≤
Eρ(x, y)
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (PI Constant) g(G, ρ)−1 = sup f :V →X f 6=const
M. Mendel (OUI)
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Non-linear spectral gaps
Eρ(x, y) ≤ g −1 (C5 , ρ) Eρ(x, y)
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (PI Constant) g(G, ρ)−1 = sup f :V →X f 6=const
M. Mendel (OUI)
Ex,y ∈V ρ(f (x), f (y )) . E(x,y )∈E ρ(f (x), f (y ))
Non-linear spectral gaps
Eρ(x, y) ≤ g −1 (C5 , ρ) Eρ(x, y) C5
K5
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (DC PI Constant) ˜ (G, ρ)−1 = g
sup f ,h:V →X
M. Mendel (OUI)
Ex,y ∈V ρ(f (x), h(y )) . E(x,y )∈E ρ(f (x), h(y ))
Non-linear spectral gaps
Eρ(x, y) ≤
Eρ(x, y)
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (DC PI Constant) ˜ (G, ρ)−1 = g
sup f ,h:V →X
Ex,y ∈V ρ(f (x), h(y )) . E(x,y )∈E ρ(f (x), h(y ))
Eρ(x, y) ≤ g˜−1 (C5 , ρ) Eρ(x, y)
K5,5
M. Mendel (OUI)
Non-linear spectral gaps
C5 ⊗ {0, 1}
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (DC PI Constant) ˜ (G, ρ)−1 = g
sup f ,h:V →X
Ex,y ∈V ρ(f (x), h(y )) . E(x,y )∈E ρ(f (x), h(y ))
Eρ(x, y) ≤ g˜−1 (C5 , ρ) Eρ(x, y)
Claim ˜ (G, ρ)−1 ∀G, ρ, g(G, ρ)−1 ≤ g ˜ (G) = 1 − λ(G), where g λ(G) = maxi≥2 |λi (G)|
M. Mendel (OUI)
Non-linear spectral gaps
K5,5
C5 ⊗ {0, 1}
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (DC PI Constant) ˜ (G, ρ)−1 = g
sup f ,h:V →X
Ex,y ∈V ρ(f (x), h(y )) . E(x,y )∈E ρ(f (x), h(y ))
Eρ(x, y) ≤ g˜−1 (C5 , ρ) Eρ(x, y)
Claim ˜ (G, ρ)−1 ∀G, ρ, g(G, ρ)−1 ≤ g ˜ (G) = 1 − λ(G), where g λ(G) = maxi≥2 |λi (G)|
C5 ⊗ {0, 1}
K5,5
Theorem ([MN] correct formulation) ∀ symmetric kernel ρ,
M. Mendel (OUI)
˜ (G ˜ (G, ρ)−1 · g ˜ (H, ρ)−2 . g z H, ρ)−1 ≤ g
Non-linear spectral gaps
Dec ’08
26 / 38
Double Cover (DC) PI Constant Definition (DC PI Constant) ˜ (G, ρ)−1 = g
sup f ,h:V →X
Ex,y ∈V ρ(f (x), h(y )) . E(x,y )∈E ρ(f (x), h(y ))
Eρ(x, y) ≤ g˜−1 (C5 , ρ) Eρ(x, y)
Claim ˜ (G, ρ)−1 ∀G, ρ, g(G, ρ)−1 ≤ g ˜ (G) = 1 − λ(G), where g λ(G) = maxi≥2 |λi (G)|
C5 ⊗ {0, 1}
K5,5
Theorem ([MN] correct formulation) ∀ symmetric kernel ρ,
˜ (G ˜ (G, ρ)−1 · g ˜ (H, ρ)−2 . g z H, ρ)−1 ≤ g
The analysis is simple, but beyond my drawing capabilities. We will instead analyze the tensor product G ⊗ H. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
26 / 38
Tensor Product
Given:
(n1 , d1 ) graph G
(n2 , d2 ) graph H
Define: K = G ⊗ H as follows:
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
27 / 38
Tensor Product
E(G ⊗ H) = {((u, x), (v , y )) : (u, v ) ∈ E(G), (x, y ) ∈ E(H)} G ⊗ H is (n1 n2 , d1 d2 ) graph. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
28 / 38
Tensor Product
E(G ⊗ H) = {((u, x), (v , y )) : (u, v ) ∈ E(G), (x, y ) ∈ E(H)} G ⊗ H is (n1 n2 , d1 d2 ) graph. M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
28 / 38
PI Const of Tensor Product Let AG the adjacency matrix of G.
Claim (Easy) AG⊗H = AG ⊗ AH . The eigenvalues of G ⊗ H are {λ(G)i λ(H)j : i ∈ [n1 ], j ∈ [n2 ]}. Therefore λ(G ⊗ H) = max{λ(G), λ(H)}. Which means ˜ (G ⊗ H)−1 = max{g ˜ (G)−1 , g ˜ (H)−1 }. g
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
29 / 38
PI Const of Tensor Product Let AG the adjacency matrix of G.
Claim (Easy) AG⊗H = AG ⊗ AH . The eigenvalues of G ⊗ H are {λ(G)i λ(H)j : i ∈ [n1 ], j ∈ [n2 ]}. Therefore λ(G ⊗ H) = max{λ(G), λ(H)}. Which means ˜ (G ⊗ H)−1 = max{g ˜ (G)−1 , g ˜ (H)−1 }. g We will prove
Proposition ∀ρ,
˜ (G ⊗ H, ρ)−1 ≤ g ˜ (G, ρ)−1 · g ˜ (H, ρ)−1 g M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
29 / 38
Analysis of Tensor Product
VK = VG × V H
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
Complete bipartite graph K30,30
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
M. Mendel (OUI)
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 ≤g
M. Mendel (OUI)
E
E
(x,y )∈EG u,v ∈VH
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 ≤g
M. Mendel (OUI)
E
E
(x,y )∈EG u,v ∈VH
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 ≤g
M. Mendel (OUI)
E
E
(x,y )∈EG u,v ∈VH
ρ(f (x, u) , h(y , v ))
Non-linear spectral gaps
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 ≤g
E
E
(x,y )∈EG u,v ∈VH
˜ (G, ρ)−1 · g ˜ (H, ρ)−1 ≤g
M. Mendel (OUI)
ρ(f (x, u) , h(y , v )) E
E
(x,y )∈EG (u,v )∈EH
Non-linear spectral gaps
ρ(f (x, u) , h(y , v ))
Dec ’08
30 / 38
Analysis of Tensor Product
E
E
u,v ∈VH x,y ∈VG
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 ≤g
E
E
(x,y )∈EG u,v ∈VH
ρ(f (x, u) , h(y , v ))
˜ (G, ρ)−1 · g ˜ (H, ρ)−1 ≤g
(x,y )∈EG (u,v )∈EH
˜ (G, ρ)−1 · g ˜ (H, ρ)−1 =g
((x,u),(y ,v ))∈EG⊗H
M. Mendel (OUI)
E
E
E
Non-linear spectral gaps
ρ(f (x, u) , h(y , v )) ρ(f (x, u) , h(y , v )) Dec ’08
30 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right? Wrong!
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right? Wrong! From this perspective the zig-zag product is the easiest step in RVW expanders — almost an “abstract nonsense".
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right? Wrong! From this perspective the zig-zag product is the easiest step in RVW expanders — almost an “abstract nonsense". (*)
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right? Wrong! From this perspective the zig-zag product is the easiest step in RVW expanders — almost an “abstract nonsense". (*) Finding geometric conditions when powering improves the PI constant and finding a base graph is much more challenging.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Wrapping up the zig-zag product Similar (but slightly more intricate) analysis proves the best known bound on g(G z H, ρ)−1 in surprising generality. The zig-zag step in RVW expanders works in a general setting. It implies the existence of expanders in the most general setting. Right? Wrong! From this perspective the zig-zag product is the easiest step in RVW expanders — almost an “abstract nonsense". (*) Finding geometric conditions when powering improves the PI constant and finding a base graph is much more challenging. We manage to find non-trivial sufficient conditions. But, we believe that weaker or other conditions are possible.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
31 / 38
Outline
1
Expanders
2
Background & Motivation
3
RVW Expanders & Zig-Zag Product
4
Discussion
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
32 / 38
Concrete Construction
For uniformly convex Banach spaces we also have a base graph and a decay of the PI const. Therefore
Theorem There are RVW expanders w.r.t uniformly convex Banach spaces.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
33 / 38
Concrete Construction
For uniformly convex Banach spaces we also have a base graph and a decay of the PI const. Therefore
Theorem There are RVW expanders w.r.t uniformly convex Banach spaces. For B-convex and CAT(0) spaces we have 2-out-3 ingredients.
Question Is there a base graph for CAT(0) spaces? Is there a uniform decay of the PI const in B-convex spaces?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
33 / 38
Open Problems We already asked:
Question (Q1) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ?
Question (Q2) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
34 / 38
Open Problems We already asked:
Question (Q1) Under what conditions on (X , ρ), there exists α : [1, ∞) → [1, ∞) such that ∀G, g(G, ρ)−1 ≤ α(g(G)−1 ) ?
Question (Q2) Under what conditions on (X , ρ) there exist d ∈ N, g −1 ≥ 1, and a family of d-regular graphs (Gn ), with g(Gn , ρ)−1 ≤ g −1 ?
Question If and when we have Q2 ∧ ¬ Q1: Can this stronger expansion property be used in other contexts?
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
34 / 38
My Pet Problem (Far away) Goal:
Question (Dichotomy) For any metric space (X , ρ). Either cX (M) = 1, ∀ finite M
∃(Mn )n , such that |Mn | = n, and cX (Mn ) = Ω(log n).
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
35 / 38
My Pet Problem (Far away) Goal:
Question (Dichotomy) For any metric space (X , ρ). Either cX (M) = 1, ∀ finite M
∃(Mn )n , such that |Mn | = n, and cX (Mn ) = Ω(log n). We do know:
Theorem (Weaker Dichotomy [M.–Naor]) For any metric space (X , ρ). Either cX (M) = 1, ∀ finite M
∃β > 0, ∃(Mn )n , such that |Mn | = n, and cX (Mn ) = Ω(logβ n).
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
35 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1
X contains all finite spaces (“cotype ∞”).
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1 2
X contains all finite spaces (“cotype ∞”). X does not (“finite cotype”), and then contains a base graph.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1 2
X contains all finite spaces (“cotype ∞”). X does not (“finite cotype”), and then contains a base graph.
Then we also need general uniform decay of the PI const .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1 2
X contains all finite spaces (“cotype ∞”). X does not (“finite cotype”), and then contains a base graph.
Then we also need general uniform decay of the PI const . If those conditions are met: On the 2nd case we can construct an expander.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1 2
X contains all finite spaces (“cotype ∞”). X does not (“finite cotype”), and then contains a base graph.
Then we also need general uniform decay of the PI const . If those conditions are met: On the 2nd case we can construct an expander. √ This approach fails for general metric spaces. Example: `∞ .
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
The Dichotomy & Expanders
An approach for proving the dichotomy: Given (X , ρ), either 1 2
X contains all finite spaces (“cotype ∞”). X does not (“finite cotype”), and then contains a base graph.
Then we also need general uniform decay of the PI const . If those conditions are met: On the 2nd case we can construct an expander. √ This approach fails for general metric spaces. Example: `∞ . It may be true for normed spaces. May be connected to open questions about metric cotype.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
36 / 38
Agenda: Non-Linear Spectral Gap
Beyond those concrete questions. An agenda emerges: “Calculus of non-linear spectral gaps" We feel that this subject may be a ground for a rich theory.
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
37 / 38
Thank You
M. Mendel (OUI)
Non-linear spectral gaps
Dec ’08
38 / 38
