Harmonic Analysis of Deep Convolutional Neural Networks

Helmut B˝olcskei

Department of Information Technology and Electrical Engineering

October 2017 joint work with Thomas Wiatowski and Philipp Grohs ImageNet CNNs win the ImageNet 2015 challenge [He et al., 2015 ]

ImageNet

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CNNs win the ImageNet 2015 challenge [He et al., 2015 ] “Carlos Kleiber conducting the Vienna Philharmonic’s New Year’s Concert 1989.”

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ] Kleiber conducting the Vienna Philharmonic’s New Year’s Concert 1989

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos .” conducting the Vienna Philharmonic’s New Year’s Concert 1989

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos Kleiber .” Vienna Philharmonic’s New Year’s Concert 1989

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos Kleiber conducting the .” New Year’s Concert 1989

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos Kleiber conducting the Vienna Philharmonic’s .” 1989

Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos Kleiber conducting the Vienna Philharmonic’s New Year’s Concert .” Describing the content of an image

CNNs generate sentences describing the content of an image [Vinyals et al., 2015 ]

“Carlos Kleiber conducting the Vienna Philharmonic’s New Year’s Concert 1989.” Feature extraction and classiﬁcation

input: f =

non-linear feature extraction

feature vector Φ(f)

linear classiﬁer

( hw, Φ(f)i > 0, ⇒ Shannon output: hw, Φ(f)i < 0, ⇒ von Neumann 1 possible with w = −1

: hw, Φ(f)i > 0 : hw, Φ(f)i < 0

kfk Φ(f) = 1

Why non-linear feature extractors?

Task: Separate two categories of data through a linear classiﬁer

: hw, fi > 0 : hw, fi < 0 : hw, Φ(f)i > 0 : hw, Φ(f)i < 0

1 possible with w = −1

kfk Φ(f) = 1

Why non-linear feature extractors?

Task: Separate two categories of data through a linear classiﬁer

: hw, fi > 0 : hw, fi < 0

not possible! Why non-linear feature extractors?

Task: Separate two categories of data through a linear classiﬁer

kfk Φ(f) = 1 1

: hw, fi > 0 : hw, Φ(f)i > 0 : hw, fi < 0 : hw, Φ(f)i < 0

1 not possible! possible with w = −1 ⇒ Linear separability in feature space!

Why non-linear feature extractors?

Task: Separate two categories of data through a linear classiﬁer

kfk Φ(f) = 1

⇒ Φ is invariant to angular component of the data Why non-linear feature extractors?

Task: Separate two categories of data through a linear classiﬁer

kfk Φ(f) = 1

⇒ Φ is invariant to angular component of the data ⇒ Linear separability in feature space! Translation invariance

Handwritten digits from the MNIST database [LeCun & Cortes, 1998 ]

Feature vector should be invariant to spatial location ⇒ translation invariance Deformation insensitivity

Feature vector should be independent of cameras (of diﬀerent resolutions), and insensitive to small acquisition jitters Scattering networks ([Mallat, 2012], [Wiatowski and HB, 2015])

||f ∗ g (k) | ∗ g (l) | ||f ∗ g (p) | ∗ g (r) | λ1 λ2 λ1 λ2

|f ∗ g (k) | |f ∗ g (p) | λ1 λ1

f feature map Scattering networks ([Mallat, 2012], [Wiatowski and HB, 2015])

||f ∗ g (k) | ∗ g (l) | ||f ∗ g (p) | ∗ g (r) | λ1 λ2 λ1 λ2

· ∗ χ3 · ∗ χ3

|f ∗ g (k) | |f ∗ g (p) | λ1 λ1

· ∗ χ · ∗ χ 2 f feature map 2

· ∗ χ1 Scattering networks ([Mallat, 2012], [Wiatowski and HB, 2015])

||f ∗ g (k) | ∗ g (l) | ||f ∗ g (p) | ∗ g (r) | λ1 λ2 λ1 λ2

· ∗ χ3 · ∗ χ3

|f ∗ g (k) | |f ∗ g (p) | λ1 λ1

· ∗ χ · ∗ χ 2 f feature map 2

· ∗ χ1 feature vector Φ(f)

General scattering networks guarantee [Wiatowski & HB, 2015 ] - (vertical) translation invariance - small deformation sensitivity essentially irrespective of ﬁlters, non-linearities, and poolings! e.g.: Learned ﬁlters

Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Filters: Semi-discrete frame Ψn := {χn} ∪ {gλn }λn∈Λn

2 2 X 2 2 2 d Ankfk2 ≤ kf ∗ χnk2 + kf ∗ gλn k ≤ Bnkfk2, ∀f ∈ L (R ) λn∈Λn e.g.: Learned ﬁlters

Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Filters: Semi-discrete frame Ψn := {χn} ∪ {gλn }λn∈Λn

2 2 X 2 2 2 d Ankfk2 ≤ kf ∗ χnk2 + kf ∗ gλn k ≤ Bnkfk2, ∀f ∈ L (R ) λn∈Λn

e.g.: Structured ﬁlters e.g.: Learned ﬁlters

Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Filters: Semi-discrete frame Ψn := {χn} ∪ {gλn }λn∈Λn

2 2 X 2 2 2 d Ankfk2 ≤ kf ∗ χnk2 + kf ∗ gλn k ≤ Bnkfk2, ∀f ∈ L (R ) λn∈Λn

e.g.: Unstructured ﬁlters Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Filters: Semi-discrete frame Ψn := {χn} ∪ {gλn }λn∈Λn

2 2 X 2 2 2 d Ankfk2 ≤ kf ∗ χnk2 + kf ∗ gλn k ≤ Bnkfk2, ∀f ∈ L (R ) λn∈Λn

e.g.: Learned ﬁlters ⇒ Satisﬁed by virtually all non-linearities used in the deep learning literature!

1 ReLU: Ln = 1; modulus: Ln = 1; logistic sigmoid: Ln = 4 ; ...

Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Non-linearities: Point-wise and Lipschitz-continuous 2 d kMn(f) − Mn(h)k2 ≤ Lnkf − hk2, ∀ f, h ∈ L (R ) Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Non-linearities: Point-wise and Lipschitz-continuous 2 d kMn(f) − Mn(h)k2 ≤ Lnkf − hk2, ∀ f, h ∈ L (R )

⇒ Satisﬁed by virtually all non-linearities used in the deep learning literature!

1 ReLU: Ln = 1; modulus: Ln = 1; logistic sigmoid: Ln = 4 ; ... ⇒ Emulates most poolings used in the deep learning literature!

e.g.: Pooling by sub-sampling Pn(f) = f with Rn = 1

Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Pooling: In continuous-time according to d/2 f 7→ Sn Pn(f)(Sn·), 2 d 2 d where Sn ≥ 1 is the pooling factor and Pn : L (R ) → L (R ) is Rn-Lipschitz-continuous Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Pooling: In continuous-time according to d/2 f 7→ Sn Pn(f)(Sn·), 2 d 2 d where Sn ≥ 1 is the pooling factor and Pn : L (R ) → L (R ) is Rn-Lipschitz-continuous

⇒ Emulates most poolings used in the deep learning literature!

e.g.: Pooling by sub-sampling Pn(f) = f with Rn = 1 Building blocks

Basic operations in the n-th network layer

g (k) λn non-lin. pool. . f .

g (r) λn non-lin. pool.

Pooling: In continuous-time according to d/2 f 7→ Sn Pn(f)(Sn·), 2 d 2 d where Sn ≥ 1 is the pooling factor and Pn : L (R ) → L (R ) is Rn-Lipschitz-continuous

⇒ Emulates most poolings used in the deep learning literature!

e.g.: Pooling by averaging Pn(f) = f ∗ φn with Rn = kφnk1 Vertical translation invariance

Theorem (Wiatowski and HB, 2015) Assume that the ﬁlters, non-linearities, and poolings satisfy

−2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N.

Let the pooling factors be Sn ≥ 1, n ∈ N. Then, n n ktk |||Φ (Ttf) − Φ (f)||| = O , S1 ...Sn 2 d d for all f ∈ L (R ), t ∈ R , n ∈ N. Vertical translation invariance

Theorem (Wiatowski and HB, 2015) Assume that the ﬁlters, non-linearities, and poolings satisfy

−2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N.

Let the pooling factors be Sn ≥ 1, n ∈ N. Then, n n ktk |||Φ (Ttf) − Φ (f)||| = O , S1 ...Sn 2 d d for all f ∈ L (R ), t ∈ R , n ∈ N.

⇒ Features become more invariant with increasing network depth! Vertical translation invariance

Theorem (Wiatowski and HB, 2015) Assume that the ﬁlters, non-linearities, and poolings satisfy

−2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N.

Let the pooling factors be Sn ≥ 1, n ∈ N. Then, n n ktk |||Φ (Ttf) − Φ (f)||| = O , S1 ...Sn 2 d d for all f ∈ L (R ), t ∈ R , n ∈ N.

Full translation invariance: If lim S1 · S2 · ... · Sn = ∞, then n→∞ n n lim |||Φ (Ttf) − Φ (f)||| = 0 n→∞ Vertical translation invariance

Theorem (Wiatowski and HB, 2015) Assume that the ﬁlters, non-linearities, and poolings satisfy

−2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N.

Let the pooling factors be Sn ≥ 1, n ∈ N. Then, n n ktk |||Φ (Ttf) − Φ (f)||| = O , S1 ...Sn 2 d d for all f ∈ L (R ), t ∈ R , n ∈ N.

The condition −2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N,

is easily satisﬁed by normalizing the ﬁlters {gλn }λn∈Λn . Vertical translation invariance

Theorem (Wiatowski and HB, 2015) Assume that the ﬁlters, non-linearities, and poolings satisfy

−2 −2 Bn ≤ min{1,Ln Rn }, ∀ n ∈ N.

Let the pooling factors be Sn ≥ 1, n ∈ N. Then, n n ktk |||Φ (Ttf) − Φ (f)||| = O , S1 ...Sn 2 d d for all f ∈ L (R ), t ∈ R , n ∈ N.

⇒ applies to general ﬁlters, non-linearities, and poolings - features become invariant in every network layer, but needs J → ∞ - applies to wavelet transform and modulus non-linearity without pooling

- features become more invariant with increasing network depth - applies to general ﬁlters, general non-linearities, and general poolings

Philosophy behind invariance results

Mallat’s “horizontal” translation invariance [Mallat, 2012 ]: 2 d d lim |||ΦW (Ttf) − ΦW (f)||| = 0, ∀f ∈ L (R ), ∀t ∈ R J→∞

“Vertical” translation invariance: n n 2 d d lim |||Φ (Ttf) − Φ (f)||| = 0, ∀f ∈ L ( ), ∀t ∈ n→∞ R R - applies to wavelet transform and modulus non-linearity without pooling

- applies to general ﬁlters, general non-linearities, and general poolings

Philosophy behind invariance results

Mallat’s “horizontal” translation invariance [Mallat, 2012 ]: 2 d d lim |||ΦW (Ttf) − ΦW (f)||| = 0, ∀f ∈ L (R ), ∀t ∈ R J→∞

- features become invariant in every network layer, but needs J → ∞

“Vertical” translation invariance: n n 2 d d lim |||Φ (Ttf) − Φ (f)||| = 0, ∀f ∈ L ( ), ∀t ∈ n→∞ R R

- features become more invariant with increasing network depth Philosophy behind invariance results

Mallat’s “horizontal” translation invariance [Mallat, 2012 ]: 2 d d lim |||ΦW (Ttf) − ΦW (f)||| = 0, ∀f ∈ L (R ), ∀t ∈ R J→∞

- features become invariant in every network layer, but needs J → ∞ - applies to wavelet transform and modulus non-linearity without pooling

“Vertical” translation invariance: n n 2 d d lim |||Φ (Ttf) − Φ (f)||| = 0, ∀f ∈ L ( ), ∀t ∈ n→∞ R R

- features become more invariant with increasing network depth - applies to general ﬁlters, general non-linearities, and general poolings Non-linear deformations

d d Non-linear deformation (Fτ f)(x) = f(x − τ(x)), where τ : R → R

For “small” τ: Non-linear deformations

d d Non-linear deformation (Fτ f)(x) = f(x − τ(x)), where τ : R → R

For “large” τ: Deformation sensitivity for signal classes

−x2 Consider (Fτ f)(x) = f(x − τ(x)) = f(x − e )

f1(x), (Fτ f1)(x)

f2(x), (Fτ f2)(x)

For given τ the amount of deformation induced 2 d can depend drastically on f ∈ L (R ) Philosophy behind deformation stability/sensitivity bounds

Mallat’s deformation stability bound [Mallat, 2012 ]: −J 2 |||ΦW (Fτ f)−ΦW (f)||| ≤ C 2 kτk∞ +JkDτk∞ +kD τk∞ kfkW , 2 d for all f ∈ HW ⊆ L (R )

- The signal class HW and the corresponding norm k · kW depend on the mother wavelet (and hence the network)

Our deformation sensitivity bound: α 2 d |||Φ(Fτ f) − Φ(f)||| ≤ CCkτk∞, ∀f ∈ C ⊆ L (R )

- The signal class C (band-limited functions, cartoon functions, or Lipschitz functions) is independent of the network Philosophy behind deformation stability/sensitivity bounds

Mallat’s deformation stability bound [Mallat, 2012 ]: −J 2 |||ΦW (Fτ f)−ΦW (f)||| ≤ C 2 kτk∞ +JkDτk∞ +kD τk∞ kfkW , 2 d for all f ∈ HW ⊆ L (R )

- Signal class description complexity implicit via norm k · kW

Our deformation sensitivity bound: α 2 d |||Φ(Fτ f) − Φ(f)||| ≤ CCkτk∞, ∀f ∈ C ⊆ L (R )

- Signal class description complexity explicit via CC - L-band-limited functions: CC = O(L) 3/2 - cartoon functions of size K: CC = O(K ) - M-Lipschitz functions CC = O(M) Philosophy behind deformation stability/sensitivity bounds

Mallat’s deformation stability bound [Mallat, 2012 ]: −J 2 |||ΦW (Fτ f)−ΦW (f)||| ≤ C 2 kτk∞ +JkDτk∞ +kD τk∞ kfkW , 2 d for all f ∈ HW ⊆ L (R )

Our deformation sensitivity bound: α 2 d |||Φ(Fτ f) − Φ(f)||| ≤ CCkτk∞, ∀f ∈ C ⊆ L (R )

- Decay rate α > 0 of the deformation error is signal-class- speciﬁc (band-limited functions: α = 1, cartoon functions: 1 α = 2 , Lipschitz functions: α = 1) Philosophy behind deformation stability/sensitivity bounds

Mallat’s deformation stability bound [Mallat, 2012 ]: −J 2 |||ΦW (Fτ f)−ΦW (f)||| ≤ C 2 kτk∞ +JkDτk∞ +kD τk∞ kfkW , 2 d for all f ∈ HW ⊆ L (R )

- The bound depends explicitly on higher order derivatives of τ

Our deformation sensitivity bound: α 2 d |||Φ(Fτ f) − Φ(f)||| ≤ CCkτk∞, ∀f ∈ C ⊆ L (R )

- The bound implicitly depends on derivative of τ via the 1 condition kDτk∞ ≤ 2d Philosophy behind deformation stability/sensitivity bounds

Mallat’s deformation stability bound [Mallat, 2012 ]: −J 2 |||ΦW (Fτ f)−ΦW (f)||| ≤ C 2 kτk∞ +JkDτk∞ +kD τk∞ kfkW , 2 d for all f ∈ HW ⊆ L (R ) - The bound is coupled to horizontal translation invariance 2 d d lim |||ΦW (Ttf) − ΦW (f)||| = 0, ∀f ∈ L (R ), ∀t ∈ R J→∞

Our deformation sensitivity bound: α 2 d |||Φ(Fτ f) − Φ(f)||| ≤ CCkτk∞, ∀f ∈ C ⊆ L (R ) - The bound is decoupled from vertical translation invariance n n 2 d d lim |||Φ (Ttf) − Φ (f)||| = 0, ∀f ∈ L ( ), ∀t ∈ n→∞ R R e.g.: Winner of the ImageNet 2015 challenge [He et al., 2015 ] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion

CNNs in a nutshell

CNNs used in practice employ potentially hundreds of layers and 10,000s of nodes! CNNs in a nutshell

CNNs used in practice employ potentially hundreds of layers and 10,000s of nodes!

e.g.: Winner of the ImageNet 2015 challenge [He et al., 2015 ] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion CNNs in a nutshell

CNNs used in practice employ potentially hundreds of layers and 10,000s of nodes!

e.g.: Winner of the ImageNet 2015 challenge [He et al., 2015 ] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion

Such depths (and breadths) pose formidable computational challenges in training and operating the network! Guarantee trivial null-space for feature extractor Φ

Specify the number of layers needed to have “most” of the input signal energy be contained in the feature vector

For a ﬁxed (possibly small) depth, design CNNs that capture “most” of the input signal energy

Topology reduction

Determine how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across layers Specify the number of layers needed to have “most” of the input signal energy be contained in the feature vector

For a ﬁxed (possibly small) depth, design CNNs that capture “most” of the input signal energy

Topology reduction

Determine how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across layers

Guarantee trivial null-space for feature extractor Φ For a ﬁxed (possibly small) depth, design CNNs that capture “most” of the input signal energy

Topology reduction

Determine how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across layers

Guarantee trivial null-space for feature extractor Φ

Specify the number of layers needed to have “most” of the input signal energy be contained in the feature vector Topology reduction

Determine how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across layers

Guarantee trivial null-space for feature extractor Φ

Specify the number of layers needed to have “most” of the input signal energy be contained in the feature vector

For a ﬁxed (possibly small) depth, design CNNs that capture “most” of the input signal energy Building blocks

Basic operations in the n-th network layer

g (k) λn | · | ↓S . f .

g (r) λn | · | ↓S

Filters: Semi-discrete frame Ψn := {χn} ∪ {gλn }λn∈Λn Non-linearity: Modulus | · | Pooling: Sub-sampling with pooling factor S ≥ 1 1 fb(ω) · gdλn (ω)

Demodulation eﬀect of modulus non-linearity

Components of feature vector given by |f ∗ gλn | ∗ χn+1

gλn (ω) χ[n+1(ω) d 1 ··· fb(ω) ··· ω Demodulation eﬀect of modulus non-linearity

Components of feature vector given by |f ∗ gλn | ∗ χn+1

gλn (ω) χ[n+1(ω) d 1 ··· fb(ω) ··· ω

1 fb(ω) · gdλn (ω)

ω Demodulation eﬀect of modulus non-linearity

Components of feature vector given by |f ∗ gλn | ∗ χn+1

gλn (ω) χ[n+1(ω) d 1 ··· fb(ω) ··· ω

1 fb(ω) · gdλn (ω)

Modulus squared:

2 |f ∗ gλn (x)| R (ω) fb·gdλn Demodulation eﬀect of modulus non-linearity

Components of feature vector given by |f ∗ gλn | ∗ χn+1

gλn (ω) χ[n+1(ω) d 1 ··· fb(ω) ··· ω

1 fb(ω) · gdλn (ω)

Φ(f) 1

V via χ[n+1 |f ∗ gλn |(ω) ω Do all non-linearities demodulate?

High-pass ﬁltered signal:

F(f ∗ gλ)

ω −2R 2R

2R |F(|f ∗ g |)| Do all non-linearitiesλ demodulate?

High-pass ﬁltered signal:

F(f ∗ gλ)

ω −2R 2R

Modulus: Yes! |F(|f ∗ gλ|)| ω ω −2R 2R −2R ... but (small) tails! 2R Do all non-linearities demodulate?

High-pass ﬁltered signal:

F(f ∗ gλ)

ω −2R 2R

Modulus squared: Yes, and sharply so!

2 |F(|f ∗ gλ| )|

ω −2R 2R ... but not Lipschitz-continuous! Do all non-linearities demodulate?

High-pass ﬁltered signal:

F(f ∗ gλ)

ω −2R 2R

Rectiﬁed linear unit: No!

|F(ReLU(f ∗ gλ))|

ω −2R 2R First goal: Quantify feature map energy decay

W2(f) ||f ∗ g (k) | ∗ g (l) | ||f ∗ g (p) | ∗ g (r) | λ1 λ2 λ1 λ2

· ∗ χ3 · ∗ χ3

W1(f) |f ∗ g (k) | |f ∗ g (p) | λ1 λ1

· ∗ χ · ∗ χ 2 f 2

· ∗ χ1 ⇒ Comprises various contructions of WH ﬁlters, wavelets, ridgelets, (α)-curvelets, shearlets

ω2

e.g.: analytic band-limited curvelets: ω1

Assumptions (on the ﬁlters)

i) Analyticity: For every ﬁlter gλn there exists a (not necessarily d canonical) orthant Hλn ⊆ R such that supp(g ) ⊆ H . cλn λn ii) High-pass: There exists δ > 0 such that X |g (ω)|2 = 0, a.e. ω ∈ B (0). cλn δ λn∈Λn Assumptions (on the ﬁlters)

⇒ Comprises various contructions of WH ﬁlters, wavelets, ridgelets, (α)-curvelets, shearlets

ω2

e.g.: analytic band-limited curvelets: ω1 2 d 0 d - square-integrable functions L (R ) = H (R ) 2 d s d - L-band-limited functions LL(R ) ⊆ H (R ), ∀L > 0, ∀s ≥ 0 s d 1 - cartoon functions [Donoho, 2001 ] CCART ⊆ H (R ), ∀s ∈ [0, 2 )

s d H (R ) contains a wide range of practically relevant signal classes

Handwritten digits from MNIST database [LeCun & Cortes, 1998 ]

Input signal classes

Sobolev functions of order s ≥ 0: n Z o s d 2 d 2 s 2 H (R ) = f ∈ L (R ) (1 + |ω| ) |fb(ω)| dω < ∞ Rd 2 d 0 d - square-integrable functions L (R ) = H (R ) 2 d s d - L-band-limited functions LL(R ) ⊆ H (R ), ∀L > 0, ∀s ≥ 0 s d 1 - cartoon functions [Donoho, 2001 ] CCART ⊆ H (R ), ∀s ∈ [0, 2 )

Handwritten digits from MNIST database [LeCun & Cortes, 1998 ]

Input signal classes

Sobolev functions of order s ≥ 0: n Z o s d 2 d 2 s 2 H (R ) = f ∈ L (R ) (1 + |ω| ) |fb(ω)| dω < ∞ Rd

s d H (R ) contains a wide range of practically relevant signal classes 2 d s d - L-band-limited functions LL(R ) ⊆ H (R ), ∀L > 0, ∀s ≥ 0 s d 1 - cartoon functions [Donoho, 2001 ] CCART ⊆ H (R ), ∀s ∈ [0, 2 )

Handwritten digits from MNIST database [LeCun & Cortes, 1998 ]

Input signal classes

Sobolev functions of order s ≥ 0: n Z o s d 2 d 2 s 2 H (R ) = f ∈ L (R ) (1 + |ω| ) |fb(ω)| dω < ∞ Rd

s d H (R ) contains a wide range of practically relevant signal classes

2 d 0 d - square-integrable functions L (R ) = H (R ) s d 1 - cartoon functions [Donoho, 2001 ] CCART ⊆ H (R ), ∀s ∈ [0, 2 )

Handwritten digits from MNIST database [LeCun & Cortes, 1998 ]

Input signal classes

Sobolev functions of order s ≥ 0: n Z o s d 2 d 2 s 2 H (R ) = f ∈ L (R ) (1 + |ω| ) |fb(ω)| dω < ∞ Rd

s d H (R ) contains a wide range of practically relevant signal classes

2 d 0 d - square-integrable functions L (R ) = H (R ) 2 d s d - L-band-limited functions LL(R ) ⊆ H (R ), ∀L > 0, ∀s ≥ 0 Input signal classes

Sobolev functions of order s ≥ 0: n Z o s d 2 d 2 s 2 H (R ) = f ∈ L (R ) (1 + |ω| ) |fb(ω)| dω < ∞ Rd

s d H (R ) contains a wide range of practically relevant signal classes

2 d 0 d - square-integrable functions L (R ) = H (R ) 2 d s d - L-band-limited functions LL(R ) ⊆ H (R ), ∀L > 0, ∀s ≥ 0 s d 1 - cartoon functions [Donoho, 2001 ] CCART ⊆ H (R ), ∀s ∈ [0, 2 )

Handwritten digits from MNIST database [LeCun & Cortes, 1998 ] ⇒ decay factor a is explicit and can be tuned via r, R

Exponential energy decay

Theorem Let the ﬁlters be wavelets with mother wavelet

supp(ψb) ⊆ [r−1, r], r > 1,

or Weyl-Heisenberg (WH) ﬁlters with prototype function

supp(gb) ⊆ [−R,R], R > 0. s d Then, for every f ∈ H (R ), there exists β > 0 such that

− n(2s+β) Wn(f) = O a 2s+β+1 ,

r2+1 1 1 where a = r2−1 in the wavelet case, and a = 2 + R in the WH case. Exponential energy decay

Theorem Let the ﬁlters be wavelets with mother wavelet

supp(ψb) ⊆ [r−1, r], r > 1,

or Weyl-Heisenberg (WH) ﬁlters with prototype function

supp(gb) ⊆ [−R,R], R > 0. s d Then, for every f ∈ H (R ), there exists β > 0 such that

− n(2s+β) Wn(f) = O a 2s+β+1 ,

r2+1 1 1 where a = r2−1 in the wavelet case, and a = 2 + R in the WH case.

⇒ decay factor a is explicit and can be tuned via r, R - β > 0 determines the decay of fb(ω) (as |ω| → ∞) according to

2 −( s + 1 + β ) |fb(ω)| ≤ µ(1 + |ω| ) 2 4 4 , ∀ |ω| ≥ L,

for some µ > 0, and L acts as an “eﬀective bandwidth” - smoother input signals (i.e., s↑) lead to faster energy decay - pooling through sub-sampling f 7→ S1/2f(S·) leads to decay a factor S

What about general ﬁlters? ⇒ polynomial energy decay!

Exponential energy decay

Exponential energy decay:

− n(2s+β) Wn(f) = O a 2s+β+1 - smoother input signals (i.e., s↑) lead to faster energy decay - pooling through sub-sampling f 7→ S1/2f(S·) leads to decay a factor S

What about general ﬁlters? ⇒ polynomial energy decay!

Exponential energy decay

Exponential energy decay:

− n(2s+β) Wn(f) = O a 2s+β+1

- β > 0 determines the decay of fb(ω) (as |ω| → ∞) according to

2 −( s + 1 + β ) |fb(ω)| ≤ µ(1 + |ω| ) 2 4 4 , ∀ |ω| ≥ L,

for some µ > 0, and L acts as an “eﬀective bandwidth” - pooling through sub-sampling f 7→ S1/2f(S·) leads to decay a factor S

What about general ﬁlters? ⇒ polynomial energy decay!

Exponential energy decay

Exponential energy decay:

− n(2s+β) Wn(f) = O a 2s+β+1

- β > 0 determines the decay of fb(ω) (as |ω| → ∞) according to

2 −( s + 1 + β ) |fb(ω)| ≤ µ(1 + |ω| ) 2 4 4 , ∀ |ω| ≥ L,

for some µ > 0, and L acts as an “eﬀective bandwidth” - smoother input signals (i.e., s↑) lead to faster energy decay What about general ﬁlters? ⇒ polynomial energy decay!

Exponential energy decay

Exponential energy decay:

− n(2s+β) Wn(f) = O a 2s+β+1

- β > 0 determines the decay of fb(ω) (as |ω| → ∞) according to

2 −( s + 1 + β ) |fb(ω)| ≤ µ(1 + |ω| ) 2 4 4 , ∀ |ω| ≥ L,

Exponential energy decay:

− n(2s+β) Wn(f) = O a 2s+β+1

- β > 0 determines the decay of fb(ω) (as |ω| → ∞) according to

2 −( s + 1 + β ) |fb(ω)| ≤ µ(1 + |ω| ) 2 4 4 , ∀ |ω| ≥ L,

What about general ﬁlters? ⇒ polynomial energy decay! Non-trivial null-space: ∃ f ∗ 6= 0 such that Φ(f ∗) = 0 ⇒ hw, Φ(f ∗)i = 0 for all w ! ⇒ these f ∗ become unclassiﬁable!

... our second goal ... trivial null-space for Φ

Why trivial null-space? Feature space

: hw, Φ(f)i > 0 : hw, Φ(f)i < 0 ... our second goal ... trivial null-space for Φ

Why trivial null-space? Feature space

: hw, Φ(f)i > 0

Φ(f ∗) : hw, Φ(f)i < 0

Non-trivial null-space: ∃ f ∗ 6= 0 such that Φ(f ∗) = 0 ⇒ hw, Φ(f ∗)i = 0 for all w ! ⇒ these f ∗ become unclassiﬁable! ... our second goal ...

Trivial null-space for feature extractor: 2 d f ∈ L (R ) | Φ(f) = 0 = 0

S∞ n Feature extractor Φ(·) = n=0 Φ (·) shall satisfy

2 2 2 2 d Akfk2 ≤ |||Φ(f)||| ≤ Bkfk2, ∀f ∈ L (R ),

for some A, B > 0. - For Parseval frames (i.e., An = Bn = 1, n ∈ N), this yields 2 2 |||Φ(f)||| = kfk2 - Connection to energy decay: n−1 2 X k 2 kfk2 = |||Φ (f)||| + Wn(f) | {z } k=0 → 0

“Energy conservation”

Theorem

For the frame upper {Bn}n∈N and frame lower bounds {An}n∈N, Q∞ Q∞ deﬁne B := n=1 max{1,Bn} and A := n=1 min{1,An}. If

0 < A ≤ B < ∞, then 2 2 2 2 d Akfk2 ≤ |||Φ(f)||| ≤ Bkfk2, ∀ f ∈ L (R ). - Connection to energy decay: n−1 2 X k 2 kfk2 = |||Φ (f)||| + Wn(f) | {z } k=0 → 0

“Energy conservation”

Theorem

For the frame upper {Bn}n∈N and frame lower bounds {An}n∈N, Q∞ Q∞ deﬁne B := n=1 max{1,Bn} and A := n=1 min{1,An}. If

0 < A ≤ B < ∞, then 2 2 2 2 d Akfk2 ≤ |||Φ(f)||| ≤ Bkfk2, ∀ f ∈ L (R ).

- For Parseval frames (i.e., An = Bn = 1, n ∈ N), this yields 2 2 |||Φ(f)||| = kfk2 “Energy conservation”

Theorem

For the frame upper {Bn}n∈N and frame lower bounds {An}n∈N, Q∞ Q∞ deﬁne B := n=1 max{1,Bn} and A := n=1 min{1,An}. If

0 < A ≤ B < ∞, then 2 2 2 2 d Akfk2 ≤ |||Φ(f)||| ≤ Bkfk2, ∀ f ∈ L (R ).

- For Parseval frames (i.e., An = Bn = 1, n ∈ N), this yields 2 2 |||Φ(f)||| = kfk2 - Connection to energy decay: n−1 2 X k 2 kfk2 = |||Φ (f)||| + Wn(f) | {z } k=0 → 0 How many layers n are needed to have at least ((1 − ε) · 100)% of the input signal energy be contained in the feature vector, i.e., n 2 X k 2 2 2 d (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2, ∀f ∈ L (R ). k=0

... and our third goal ...

For a given CNN, specify the number of layers needed to capture “most” of the input signal energy ... and our third goal ...

For a given CNN, specify the number of layers needed to capture “most” of the input signal energy

How many layers n are needed to have at least ((1 − ε) · 100)% of the input signal energy be contained in the feature vector, i.e., n 2 X k 2 2 2 d (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2, ∀f ∈ L (R ). k=0 Number of layers needed

Theorem Let the frame bounds satisfy An = Bn = 1, n ∈ N. Let the input signal f be L-band-limited, and let ε ∈ (0, 1). If

L n ≥ log √ , a (1 − 1 − ε )

then n 2 X k 2 2 (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2. k=0 Number of layers needed

Theorem Let the frame bounds satisfy An = Bn = 1, n ∈ N. Let the input signal f be L-band-limited, and let ε ∈ (0, 1). If

L n ≥ log √ , a (1 − 1 − ε )

then n 2 X k 2 2 (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2. k=0

Sn k ⇒ also guarantees trivial null-space for k=0 Φ (f) - similar estimates for Sobolev input signals and for general ﬁlters (polynomial decay!)

Number of layers needed

Theorem Let the frame bounds satisfy An = Bn = 1, n ∈ N. Let the input signal f be L-band-limited, and let ε ∈ (0, 1). If

L n ≥ log √ , a (1 − 1 − ε )

then n 2 X k 2 2 (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2. k=0

- lower bound depends on - description complexity of input signals (i.e., bandwidth L) r2+1 1 1 - decay factor (wavelets a = r2−1 , WH ﬁlters a = 2 + R ) Number of layers needed

Theorem Let the frame bounds satisfy An = Bn = 1, n ∈ N. Let the input signal f be L-band-limited, and let ε ∈ (0, 1). If

L n ≥ log √ , a (1 − 1 − ε )

then n 2 X k 2 2 (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2. k=0

- lower bound depends on - description complexity of input signals (i.e., bandwidth L) r2+1 1 1 - decay factor (wavelets a = r2−1 , WH ﬁlters a = 2 + R ) - similar estimates for Sobolev input signals and for general ﬁlters (polynomial decay!) Recall: Winner of the ImageNet 2015 challenge [He et al., 2015] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion

Number of layers needed

Numerical example for bandwidth L = 1: (1 − ε) 0.25 0.5 0.75 0.9 0.95 0.99 wavelets (r = 2) 2 3 4 6 8 11 WH ﬁlters (R = 1) 2 4 5 8 10 14 general ﬁlters 2 3 7 19 39 199 Recall: Winner of the ImageNet 2015 challenge [He et al., 2015] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion

Number of layers needed

Numerical example for bandwidth L = 1: (1 − ε) 0.25 0.5 0.75 0.9 0.95 0.99 wavelets (r = 2) 2 3 4 6 8 11 WH ﬁlters (R = 1) 2 4 5 8 10 14 general ﬁlters 2 3 7 19 39 199 Number of layers needed

Numerical example for bandwidth L = 1: (1 − ε) 0.25 0.5 0.75 0.9 0.95 0.99 wavelets (r = 2) 2 3 4 6 8 11 WH ﬁlters (R = 1) 2 4 5 8 10 14 general ﬁlters 2 3 7 19 39 199

Recall: Winner of the ImageNet 2015 challenge [He et al., 2015] - Network depth: 152 layers - average # of nodes per layer: 472 - # of FLOPS for a single forward pass: 11.3 billion ... our fourth and last goal ...

For a ﬁxed (possibly small) depth N, design scattering networks that capture “most” of the input signal energy ... our fourth and last goal ...

For a ﬁxed (possibly small) depth N, design scattering networks that capture “most” of the input signal energy

Recall: Let the ﬁlters be wavelets with mother wavelet

supp(ψb) ⊆ [r−1, r], r > 1,

or Weyl-Heisenberg ﬁlters with prototype function

supp(gb) ⊆ [−R,R], R > 0. ... our fourth and last goal ...

For a ﬁxed (possibly small) depth N, design scattering networks that capture “most” of the input signal energy

For ﬁxed depth N, want to choose r in the wavelet and R in the WH case so that

N 2 X k 2 2 2 d (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2, ∀f ∈ L (R ). k=0 Depth-constrained networks

Theorem Let the frame bounds satisfy An = Bn = 1, n ∈ N. Let the input signal f be L-band-limited, and ﬁx ε ∈ (0, 1) and N ∈ N. If, in the wavelet case, rκ + 1 1 < r ≤ , κ − 1 or, in the WH case, s 1 0 < R ≤ 1 , κ − 2 1 N where κ := √L , then (1− 1−ε ) N 2 X k 2 2 (1 − ε)kfk2 ≤ |||Φ (f)||| ≤ kfk2. k=0 Smaller depth N ⇒ smaller “bandwidth” r of mother wavelet

⇒ larger number of wavelets (O(logr(L))) to cover the spectral support [−L, L] of input signal ⇒ larger number of filters in the first layer ⇒ depth-width tradeoff

Depth-width tradeoﬀ

Spectral supports of wavelet ﬁlters:

g g g ψb b1 b2 b3 1

ω 1 1 r 2 3 L r r r ⇒ larger number of wavelets (O(logr(L))) to cover the spectral support [−L, L] of input signal ⇒ larger number of filters in the first layer ⇒ depth-width tradeoff

Depth-width tradeoﬀ

Spectral supports of wavelet ﬁlters:

g g g ψb b1 b2 b3 1

ω 1 1 r 2 3 L r r r

Smaller depth N ⇒ smaller “bandwidth” r of mother wavelet ⇒ larger number of filters in the first layer ⇒ depth-width tradeoff

Depth-width tradeoﬀ

Spectral supports of wavelet ﬁlters:

g g g ψb b1 b2 b3 1

ω 1 1 r 2 3 L r r r

Smaller depth N ⇒ smaller “bandwidth” r of mother wavelet

⇒ larger number of wavelets (O(logr(L))) to cover the spectral support [−L, L] of input signal ⇒ depth-width tradeoﬀ

Depth-width tradeoﬀ

Spectral supports of wavelet ﬁlters:

g g g ψb b1 b2 b3 1

ω 1 1 r 2 3 L r r r

Smaller depth N ⇒ smaller “bandwidth” r of mother wavelet

⇒ larger number of wavelets (O(logr(L))) to cover the spectral support [−L, L] of input signal ⇒ larger number of filters in the first layer Depth-width tradeoff

Spectral supports of wavelet ﬁlters:

g g g ψb b1 b2 b3 1

ω 1 1 r 2 3 L r r r

Smaller depth N ⇒ smaller “bandwidth” r of mother wavelet

⇒ larger number of wavelets (O(logr(L))) to cover the spectral support [−L, L] of input signal ⇒ larger number of filters in the first layer ⇒ depth-width tradeoff Yours truly Experiment: Handwritten digit classification

- Dataset: MNIST database of handwritten digits [LeCun & Cortes, 1998]; 60,000 training and 10,000 test images - Φ-network: D = 3 layers; same filters, non-linearities, and pooling operators in all layers - Classifier: SVM with radial basis function kernel [Vapnik, 1995 ] - Dimensionality reduction: Supervised orthogonal least squares scheme [Chen et al., 1991] - modulus and ReLU perform better than tanh and LogSig - results with pooling (S = 2) are competitive with those without pooling, at significanly lower computational cost - state-of-the-art: 0.43 [Bruna and Mallat, 2013] - similar feature extraction network with directional, non-separable wavelets and no pooling - significantly higher computational complexity

Experiment: Handwritten digit classiﬁcation

Classification error in percent: Haar wavelet Bi-orthogonal wavelet abs ReLU tanh LogSig abs ReLU tanh LogSig n.p. 0.57 0.57 1.35 1.49 0.51 0.57 1.12 1.22 sub. 0.69 0.66 1.25 1.46 0.61 0.61 1.20 1.18 max. 0.58 0.65 0.75 0.74 0.52 0.64 0.78 0.73 avg. 0.55 0.60 1.27 1.35 0.58 0.59 1.07 1.26 - results with pooling (S = 2) are competitive with those without pooling, at significanly lower computational cost - state-of-the-art: 0.43 [Bruna and Mallat, 2013] - similar feature extraction network with directional, non-separable wavelets and no pooling - significantly higher computational complexity

Experiment: Handwritten digit classiﬁcation

- modulus and ReLU perform better than tanh and LogSig - state-of-the-art: 0.43 [Bruna and Mallat, 2013] - similar feature extraction network with directional, non-separable wavelets and no pooling - signiﬁcantly higher computational complexity

Experiment: Handwritten digit classiﬁcation

- modulus and ReLU perform better than tanh and LogSig - results with pooling (S = 2) are competitive with those without pooling, at signiﬁcanly lower computational cost Experiment: Handwritten digit classiﬁcation

- modulus and ReLU perform better than tanh and LogSig - results with pooling (S = 2) are competitive with those without pooling, at signiﬁcanly lower computational cost - state-of-the-art: 0.43 [Bruna and Mallat, 2013] - similar feature extraction network with directional, non-separable wavelets and no pooling - signiﬁcantly higher computational complexity - vanishing moments condition on the mother wavelet 2 - applies to 1-D real-valued band-limited input signals f ∈ L (R)

Energy decay: Related work

[Waldspurger, 2017 ]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspeciﬁed a > 1. - 1-D wavelet ﬁlters - every network layer equipped with the same set of wavelets 2 - applies to 1-D real-valued band-limited input signals f ∈ L (R)

Energy decay: Related work

[Waldspurger, 2017 ]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspeciﬁed a > 1. - 1-D wavelet ﬁlters - every network layer equipped with the same set of wavelets - vanishing moments condition on the mother wavelet Energy decay: Related work

[Waldspurger, 2017 ]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspecified a > 1. - 1-D wavelet filters - every network layer equipped with the same set of wavelets - vanishing moments condition on the mother wavelet 2 - applies to 1-D real-valued band-limited input signals f ∈ L (R) - analyticity and vanishing moments conditions on the filters - applies to d-dimensional complex-valued input signals 2 d f ∈ L (R )

Energy decay: Related work

[Czaja and Li, 2016]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspecified a > 1. - d-dimensional uniform covering filters (similar to Weyl- Heisenberg filters), but does not cover multi-scale filters (e.g. wavelets, ridgedelets, curvelets etc.) - every network layer equipped with the same set of filters - applies to d-dimensional complex-valued input signals 2 d f ∈ L (R )

Energy decay: Related work

[Czaja and Li, 2016]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspecified a > 1. - d-dimensional uniform covering filters (similar to Weyl- Heisenberg filters), but does not cover multi-scale filters (e.g. wavelets, ridgedelets, curvelets etc.) - every network layer equipped with the same set of filters - analyticity and vanishing moments conditions on the filters Energy decay: Related work

[Czaja and Li, 2016]: Exponential energy decay

−n Wn(f) = O(a ),

for some unspecified a > 1. - d-dimensional uniform covering filters (similar to Weyl- Heisenberg filters), but does not cover multi-scale filters (e.g. wavelets, ridgedelets, curvelets etc.) - every network layer equipped with the same set of filters - analyticity and vanishing moments conditions on the filters - applies to d-dimensional complex-valued input signals 2 d f ∈ L (R )