Sparse Fourier Transforms Eric Price UT Austin Eric Price Sparse - - PowerPoint PPT Presentation

▶
sparse fourier transforms
SMART_READER_LITE
LIVE PREVIEW

Sparse Fourier Transforms Eric Price UT Austin Eric Price Sparse - - PowerPoint PPT Presentation

Jul 18, 2023 129 likes •1.94k views

Sparse Fourier Transforms Eric Price UT Austin Eric Price Sparse Fourier Transforms 1 / 36 The Fourier Transform Conversion between time and frequency domains Frequency Domain Time Domain Fourier Transform Displacement of Air Concert A

slide-1
SLIDE 1

Sparse Fourier Transforms

Eric Price

UT Austin

Eric Price Sparse Fourier Transforms 1 / 36

slide-2
SLIDE 2

The Fourier Transform

Conversion between time and frequency domains

Time Domain Frequency Domain Fourier Transform Displacement of Air Concert A

Eric Price Sparse Fourier Transforms 2 / 36

slide-3
SLIDE 3

The Fourier Transform is Ubiquitous

Audio Video Medical Imaging Radar GPS Oil Exploration

Eric Price Sparse Fourier Transforms 3 / 36

slide-4
SLIDE 4

Computing the Discrete Fourier Transform

How to compute x = Fx?

Eric Price Sparse Fourier Transforms 4 / 36

slide-5
SLIDE 5

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2).

Eric Price Sparse Fourier Transforms 4 / 36

slide-6
SLIDE 6

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965]

Eric Price Sparse Fourier Transforms 4 / 36

slide-7
SLIDE 7

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965] [T]he method greatly reduces the tediousness of mechanical calculations. – Carl Friedrich Gauss, 1805

Eric Price Sparse Fourier Transforms 4 / 36

slide-8
SLIDE 8

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965] [T]he method greatly reduces the tediousness of mechanical calculations. – Carl Friedrich Gauss, 1805 By hand: 22n log n seconds. [Danielson-Lanczos, 1942]

Eric Price Sparse Fourier Transforms 4 / 36

slide-9
SLIDE 9

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965] [T]he method greatly reduces the tediousness of mechanical calculations. – Carl Friedrich Gauss, 1805 By hand: 22n log n seconds. [Danielson-Lanczos, 1942] Can we do better?

Eric Price Sparse Fourier Transforms 4 / 36

slide-10
SLIDE 10

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965] [T]he method greatly reduces the tediousness of mechanical calculations. – Carl Friedrich Gauss, 1805 By hand: 22n log n seconds. [Danielson-Lanczos, 1942] Can we do much better?

Eric Price Sparse Fourier Transforms 4 / 36

slide-11
SLIDE 11

Computing the Discrete Fourier Transform

How to compute x = Fx? Naive multiplication: O(n2). Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965] [T]he method greatly reduces the tediousness of mechanical calculations. – Carl Friedrich Gauss, 1805 By hand: 22n log n seconds. [Danielson-Lanczos, 1942] Can we do much better? When can we compute the Fourier Transform in sublinear time?

Eric Price Sparse Fourier Transforms 4 / 36

slide-12
SLIDE 12

Idea: Leverage Sparsity

Often the Fourier transform is dominated by a small number of peaks: Time Signal Frequency (Exactly sparse) Frequency (Approximately sparse)

Eric Price Sparse Fourier Transforms 5 / 36

slide-13
SLIDE 13

Idea: Leverage Sparsity

Often the Fourier transform is dominated by a small number of peaks: Time Signal Frequency (Exactly sparse) Frequency (Approximately sparse) Sparsity is common:

Audio Video Medical Imaging Radar GPS Oil Exploration

Eric Price Sparse Fourier Transforms 5 / 36

slide-14
SLIDE 14

Idea: Leverage Sparsity

Often the Fourier transform is dominated by a small number of peaks: Time Signal Frequency (Exactly sparse) Frequency (Approximately sparse) Sparsity is common:

Audio Video Medical Imaging Radar GPS Oil Exploration

Goal of this talk: sparse Fourier transforms

Faster Fourier Transform on sparse data.

Eric Price Sparse Fourier Transforms 5 / 36

slide-15
SLIDE 15

Recent Theory and Applied Work

Sparse Fourier Transform in the Discrete Setting

◮ Gilbert-Guha-Indyk-Muthukrishnan-Strauss, 02 ◮ Gilbert-Muthukrishnan-Strauss, 05 ◮ Hassanieh-Indyk-Katabi-Price, 12 ◮ Indyk-Kapralov, 14 Eric Price Sparse Fourier Transforms 6 / 36

slide-16
SLIDE 16

Recent Theory and Applied Work

Sparse Fourier Transform in the Discrete Setting

◮ Gilbert-Guha-Indyk-Muthukrishnan-Strauss, 02 ◮ Gilbert-Muthukrishnan-Strauss, 05 ◮ Hassanieh-Indyk-Katabi-Price, 12 ◮ Indyk-Kapralov, 14

Sparse Fourier Transform in the Continuous Setting

◮ Boufounos-Cevher-Gilbert-Li-Strauss, 12 ◮ Price-Song, 15 Eric Price Sparse Fourier Transforms 6 / 36

slide-17
SLIDE 17

Recent Theory and Applied Work

Sparse Fourier Transform in the Discrete Setting

◮ Gilbert-Guha-Indyk-Muthukrishnan-Strauss, 02 ◮ Gilbert-Muthukrishnan-Strauss, 05 ◮ Hassanieh-Indyk-Katabi-Price, 12 ◮ Indyk-Kapralov, 14

Sparse Fourier Transform in the Continuous Setting

◮ Boufounos-Cevher-Gilbert-Li-Strauss, 12 ◮ Price-Song, 15

Applications

Faster GPS ... Fourier ... Hassanieh et al. MOBICOM’12 ... Fourier ... Chip ... Abari et al. ISSCC’12 ... Chemical ... Imaging ... Andronesi et al. ENC’14 Light ... Continuous Fourier... Shi et al. SIGGRAPH’15 Eric Price Sparse Fourier Transforms 6 / 36

slide-18
SLIDE 18

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj

Eric Price Sparse Fourier Transforms 7 / 36

slide-19
SLIDE 19

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj 2d Fourier Transform: x ∈ Cn1×n2, ωi = e2πi/ni, want

  • xi1,i2 =

n1

  • j1=1

n2

  • j2=1

ωi1j1

1 ωi2j2 2 xj1,j2

Eric Price Sparse Fourier Transforms 7 / 36

slide-20
SLIDE 20

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj 2d Fourier Transform: x ∈ Cn1×n2, ωi = e2πi/ni, want

  • xi1,i2 =

n1

  • j1=1

n2

  • j2=1

ωi1j1

1 ωi2j2 2 xj1,j2

Eric Price Sparse Fourier Transforms 7 / 36

slide-21
SLIDE 21

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj 2d Fourier Transform: x ∈ Cn1×n2, ωi = e2πi/ni, want

  • xi1,i2 =

n1

  • j1=1

n2

  • j2=1

ωi1j1

1 ωi2j2 2 xj1,j2

Eric Price Sparse Fourier Transforms 7 / 36

slide-22
SLIDE 22

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj 2d Fourier Transform: x ∈ Cn1×n2, ωi = e2πi/ni, want

  • xi1,i2 =

n1

  • j1=1

n2

  • j2=1

ωi1j1

1 ωi2j2 2 xj1,j2

◮ If n1, n2 are relatively prime, equivalent to 1d transform of Cn1n2 Eric Price Sparse Fourier Transforms 7 / 36

slide-23
SLIDE 23

Kinds of discrete Fourier transform

1d Fourier transform: x ∈ Cn, ω = e2πi/n, want

  • xi =

n

  • j=1

ωijxj 2d Fourier Transform: x ∈ Cn1×n2, ωi = e2πi/ni, want

  • xi1,i2 =

n1

  • j1=1

n2

  • j2=1

ωi1j1

1 ωi2j2 2 xj1,j2

◮ If n1, n2 are relatively prime, equivalent to 1d transform of Cn1n2

Hadamard transform: x ∈ C2×2×···×2:

  • xi =

n

  • j

(−1)i,jxj

Eric Price Sparse Fourier Transforms 7 / 36

slide-24
SLIDE 24

Generic Algorithm Outline

Goal: given access to x, compute x ≈ x

◮ Exact case:

x is k-sparse, x = x (maybe to log n bits of precision)

Eric Price Sparse Fourier Transforms 8 / 36

slide-25
SLIDE 25

Generic Algorithm Outline

Goal: given access to x, compute x ≈ x

◮ Exact case:

x is k-sparse, x = x (maybe to log n bits of precision)

◮ Approximate case:

x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2

Eric Price Sparse Fourier Transforms 8 / 36

slide-26
SLIDE 26

Generic Algorithm Outline

Goal: given access to x, compute x ≈ x

◮ Exact case:

x is k-sparse, x = x (maybe to log n bits of precision)

◮ Approximate case:

x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2

◮ With “good” probability. Eric Price Sparse Fourier Transforms 8 / 36

slide-27
SLIDE 27

Generic Algorithm Outline

Goal: given access to x, compute x ≈ x

◮ Exact case:

x is k-sparse, x = x (maybe to log n bits of precision)

◮ Approximate case:

x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2

◮ With “good” probability. 1

Algorithm for k = 1 (exact or approximate)

Eric Price Sparse Fourier Transforms 8 / 36

slide-28
SLIDE 28

Generic Algorithm Outline

Goal: given access to x, compute x ≈ x

◮ Exact case:

x is k-sparse, x = x (maybe to log n bits of precision)

◮ Approximate case:

x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2

◮ With “good” probability. 1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

Eric Price Sparse Fourier Transforms 8 / 36

slide-29
SLIDE 29

Generic Algorithm Outline

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

Eric Price Sparse Fourier Transforms 8 / 36

slide-30
SLIDE 30

Generic Algorithm Outline

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

Eric Price Sparse Fourier Transforms 8 / 36

slide-31
SLIDE 31

Generic Algorithm Outline

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

◮ Split

x into O(k) “random” parts

Eric Price Sparse Fourier Transforms 8 / 36

slide-32
SLIDE 32

Generic Algorithm Outline

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

◮ Split

x into O(k) “random” parts

◮ Can sample time domain of the parts. Eric Price Sparse Fourier Transforms 8 / 36

slide-33
SLIDE 33

Generic Algorithm Outline

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

◮ Split

x into O(k) “random” parts

◮ Can sample time domain of the parts. ⋆ O(k log k) time to get one sample from each of the k parts. Eric Price Sparse Fourier Transforms 8 / 36

slide-34
SLIDE 34

Generic Algorithm Outline

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

1

Algorithm for k = 1 (exact or approximate)

2

Method to reduce to k = 1 case

◮ Split

x into O(k) “random” parts

◮ Can sample time domain of the parts. ⋆ O(k log k) time to get one sample from each of the k parts. 3

Finds “most” of signal; repeat on residual

Eric Price Sparse Fourier Transforms 8 / 36

slide-35
SLIDE 35

Talk Outline

1

Algorithm for k = 1

Eric Price Sparse Fourier Transforms 9 / 36

slide-36
SLIDE 36

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

Eric Price Sparse Fourier Transforms 9 / 36

slide-37
SLIDE 37

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

Eric Price Sparse Fourier Transforms 9 / 36

slide-38
SLIDE 38

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

4

Continuous setting

Eric Price Sparse Fourier Transforms 9 / 36

slide-39
SLIDE 39

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

4

Continuous setting

Eric Price Sparse Fourier Transforms 10 / 36

slide-40
SLIDE 40

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Eric Price Sparse Fourier Transforms 11 / 36

slide-41
SLIDE 41

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t).

Eric Price Sparse Fourier Transforms 11 / 36

slide-42
SLIDE 42

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t). x0 = a

Eric Price Sparse Fourier Transforms 11 / 36

slide-43
SLIDE 43

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t). x0 = a x1 = aωt

Eric Price Sparse Fourier Transforms 11 / 36

slide-44
SLIDE 44

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t). x0 = a x1 = aωt x1/x0 = ωt = ⇒ t.

Eric Price Sparse Fourier Transforms 11 / 36

slide-45
SLIDE 45

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t). x0 = a x1 = aωt x1/x0 = ωt = ⇒ t.

  • Eric Price

Sparse Fourier Transforms 11 / 36

slide-46
SLIDE 46

Algorithm for k = 1: one dimension, exact case

  • x:

t a

Lemma

We can compute a 1-sparse x in O(1) time.

  • xi =

a if i = t

  • therwise

Then x = (a, aωt, aω2t, aω3t, . . . , aω(n−1)t). x0 = a x1 = aωt x1/x0 = ωt = ⇒ t.

  • (Related to OFDM, Prony’s method, matrix pencil.)

Eric Price Sparse Fourier Transforms 11 / 36

slide-47
SLIDE 47

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

Eric Price Sparse Fourier Transforms 12 / 36

slide-48
SLIDE 48

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

x1/x0 = ωt

With exact sparsity: log n bits in a single measurement.

Eric Price Sparse Fourier Transforms 12 / 36

slide-49
SLIDE 49

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

x1/x0 = ωt + noise

With exact sparsity: log n bits in a single measurement. With noise: only constant number of useful bits.

Eric Price Sparse Fourier Transforms 12 / 36

slide-50
SLIDE 50

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

x1/x0 = ωt + noise

With exact sparsity: log n bits in a single measurement. With noise: only constant number of useful bits. Choose Θ(log n) time shifts c to recover i.

Eric Price Sparse Fourier Transforms 12 / 36

slide-51
SLIDE 51

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

xc2/x0 = ωc2t + noise

With exact sparsity: log n bits in a single measurement. With noise: only constant number of useful bits. Choose Θ(log n) time shifts c to recover i.

Eric Price Sparse Fourier Transforms 12 / 36

slide-52
SLIDE 52

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

xc3/x0 = ωc3t + noise

With exact sparsity: log n bits in a single measurement. With noise: only constant number of useful bits. Choose Θ(log n) time shifts c to recover i.

Eric Price Sparse Fourier Transforms 12 / 36

slide-53
SLIDE 53

Algorithm for k = 1: one dimension, approximate case

Lemma

Suppose x is approximately 1-sparse: | xt|/ x2 90%. Then we can recover it with O(log n) samples and O(log2 n) time.

xc3/x0 = ωc3t + noise

With exact sparsity: log n bits in a single measurement. With noise: only constant number of useful bits. Choose Θ(log n) time shifts c to recover i. Error correcting code with efficient recovery = ⇒ lemma.

  • Eric Price

Sparse Fourier Transforms 12 / 36

slide-54
SLIDE 54

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

4

Continuous setting

Eric Price Sparse Fourier Transforms 13 / 36

slide-55
SLIDE 55

Algorithm for general k

Reduce general k to k = 1.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-56
SLIDE 56

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-57
SLIDE 57

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-58
SLIDE 58

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-59
SLIDE 59

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-60
SLIDE 60

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-61
SLIDE 61

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-62
SLIDE 62

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-63
SLIDE 63

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-64
SLIDE 64

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-65
SLIDE 65

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket. Random permutation

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-66
SLIDE 66

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket. Random permutation

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-67
SLIDE 67

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket. Random permutation

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 14 / 36

slide-68
SLIDE 68

Algorithm for general k

Reduce general k to k = 1. “Filters”: partition frequencies into O(k) buckets.

◮ Sample from time domain of each

bucket with O(log n) overhead.

◮ Recovered by k = 1 algorithm

Most frequencies alone in bucket. Random permutation

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Recovers most of x:

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse.

Eric Price Sparse Fourier Transforms 14 / 36

slide-69
SLIDE 69

Going from finding most coordinates to finding all

  • x

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse.

Eric Price Sparse Fourier Transforms 15 / 36

slide-70
SLIDE 70

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse.

Eric Price Sparse Fourier Transforms 15 / 36

slide-71
SLIDE 71

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Eric Price Sparse Fourier Transforms 15 / 36

slide-72
SLIDE 72

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Eric Price Sparse Fourier Transforms 15 / 36

slide-73
SLIDE 73

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Eric Price Sparse Fourier Transforms 15 / 36

slide-74
SLIDE 74

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Eric Price Sparse Fourier Transforms 15 / 36

slide-75
SLIDE 75

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Theorem

We can compute x in O(k log n) expected time.

Eric Price Sparse Fourier Transforms 15 / 36

slide-76
SLIDE 76

Going from finding most coordinates to finding all

  • x −

x ′

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

Partial k-sparse recovery x

  • x ′

Lemma (Partial sparse recovery)

In O(k log n) expected time, we can compute an estimate x ′ such that

  • x −

x ′ is k/2-sparse. Repeat, k → k/2 → k/4 → · · ·

Theorem

We can compute x in O(k log n) expected time.

Eric Price Sparse Fourier Transforms 15 / 36

slide-77
SLIDE 77

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

4

Continuous setting

Eric Price Sparse Fourier Transforms 16 / 36

slide-78
SLIDE 78

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x

slide-79
SLIDE 79

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x

slide-80
SLIDE 80

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x n-dimensional DFT of first k terms: O(n log n) x · rect → x ∗ sinc.

slide-81
SLIDE 81

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x n-dimensional DFT of first k terms: O(n log n) x · rect → x ∗ sinc.

slide-82
SLIDE 82

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x n-dimensional DFT of first k terms: O(n log n) x · rect → x ∗ sinc. k-dimensional DFT of first k terms: O(B log B) alias(x · rect) → subsample( x ∗ sinc).

slide-83
SLIDE 83

How can you hope for sublinear time?

Time Frequency

× = ∗ =

Eric Price Sparse Fourier Transforms 17 / 36

n-dimensional DFT: O(n log n) x → x n-dimensional DFT of first k terms: O(n log n) x · rect → x ∗ sinc. k-dimensional DFT of first k terms: O(B log B) alias(x · rect) → subsample( x ∗ sinc).

slide-84
SLIDE 84

Use a better filter

GMS05, HIKP12, IKP14, IK14

Filter (time): Gaussian · sinc Filter (frequency): Gaussian * rectangle

Filter: sparse F for which F is large on an interval.

Eric Price Sparse Fourier Transforms 18 / 36

slide-85
SLIDE 85

Use a better filter

GMS05, HIKP12, IKP14, IK14

Filter (time): Gaussian · sinc Filter (frequency): Gaussian * rectangle

Filter: sparse F for which F is large on an interval. We can permute the frequencies: x ′

i = xσi =

⇒ xi = xσ−1i

Eric Price Sparse Fourier Transforms 18 / 36

slide-86
SLIDE 86

Use a better filter

GMS05, HIKP12, IKP14, IK14

Filter (time): Gaussian · sinc Filter (frequency): Gaussian * rectangle

Filter: sparse F for which F is large on an interval. We can permute the frequencies: x ′

i = xσi =

⇒ xi = xσ−1i Allows us to convert worst case to random case.

Eric Price Sparse Fourier Transforms 18 / 36

slide-87
SLIDE 87

Algorithm for exactly sparse signals

Original signal x Goal ˆ x

Eric Price Sparse Fourier Transforms 19 / 36

slide-88
SLIDE 88

Algorithm for exactly sparse signals

Computed F ·x Filtered signal c F ∗c x

Eric Price Sparse Fourier Transforms 19 / 36

slide-89
SLIDE 89

Algorithm for exactly sparse signals

F ·x aliased to k terms Filtered signal c F ∗c x

Eric Price Sparse Fourier Transforms 19 / 36

slide-90
SLIDE 90

Algorithm for exactly sparse signals

F ·x aliased to k terms Computed samples of c F ∗c x

Eric Price Sparse Fourier Transforms 19 / 36

slide-91
SLIDE 91

Algorithm for exactly sparse signals

F ·x aliased to k terms Computed samples of c F ∗c x

Eric Price Sparse Fourier Transforms 19 / 36

slide-92
SLIDE 92

Algorithm for exactly sparse signals

F ·x aliased to k terms Knowledge about ˆ x

Eric Price Sparse Fourier Transforms 19 / 36

slide-93
SLIDE 93

Algorithm for exactly sparse signals

F ·x aliased to k terms Knowledge about ˆ x

Eric Price Sparse Fourier Transforms 19 / 36

slide-94
SLIDE 94

Algorithm for exactly sparse signals

F ·x aliased to k terms Knowledge about ˆ x

Lemma

If t is isolated in its bucket and in the “super-pass” region, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time.

Eric Price Sparse Fourier Transforms 19 / 36

slide-95
SLIDE 95

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time.

Eric Price Sparse Fourier Transforms 20 / 36

slide-96
SLIDE 96

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt

Eric Price Sparse Fourier Transforms 20 / 36

slide-97
SLIDE 97

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt = ⇒ can compute t.

Eric Price Sparse Fourier Transforms 20 / 36

slide-98
SLIDE 98

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt = ⇒ can compute t.

◮ Just like our 1-sparse recovery algorithm, x1/x0 = ωt. Eric Price Sparse Fourier Transforms 20 / 36

slide-99
SLIDE 99

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt = ⇒ can compute t.

◮ Just like our 1-sparse recovery algorithm, x1/x0 = ωt.

Gives partial sparse recovery: x ′ such that x − x ′ is k/2-sparse.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Eric Price Sparse Fourier Transforms 20 / 36

slide-100
SLIDE 100

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt = ⇒ can compute t.

◮ Just like our 1-sparse recovery algorithm, x1/x0 = ωt.

Gives partial sparse recovery: x ′ such that x − x ′ is k/2-sparse.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Repeat k → k/2 → k/4 → · · ·

Eric Price Sparse Fourier Transforms 20 / 36

slide-101
SLIDE 101

Algorithm

Lemma

For most t, the value b we compute for its bucket satisfies b = xt. Computing the b for all O(k) buckets takes O(k log n) time. Time-shift x by one and repeat: b′ = xtωt. Divide to get b′/b = ωt = ⇒ can compute t.

◮ Just like our 1-sparse recovery algorithm, x1/x0 = ωt.

Gives partial sparse recovery: x ′ such that x − x ′ is k/2-sparse.

Permute Filters O(k) 1-sparse recovery 1-sparse recovery 1-sparse recovery 1-sparse recovery

x

  • x ′

Repeat k → k/2 → k/4 → · · · O(k log n) time sparse Fourier transform.

  • Eric Price

Sparse Fourier Transforms 20 / 36

slide-102
SLIDE 102

Summary (DFT setting)

Given access to x for which x is sparse.

Eric Price Sparse Fourier Transforms 21 / 36

slide-103
SLIDE 103

Summary (DFT setting)

Given access to x for which x is sparse. Recover x such that x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2

Eric Price Sparse Fourier Transforms 21 / 36

slide-104
SLIDE 104

Summary (DFT setting)

Given access to x for which x is sparse. Recover x such that x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2 “Optimal” is O(k log(n/k)) samples and O(k log(n/k) log n) time

Eric Price Sparse Fourier Transforms 21 / 36

slide-105
SLIDE 105

Summary (DFT setting)

Given access to x for which x is sparse. Recover x such that x − x2 (1 + ǫ) min

k-sparse xk

  • x −

xk2 “Optimal” is O(k log(n/k)) samples and O(k log(n/k) log n) time

◮ Optimal samples [IK ’14] OR optimal time [HIKP ’12] OR

logc log n-competitive mixture [IKP ’14].

Eric Price Sparse Fourier Transforms 21 / 36

slide-106
SLIDE 106

Talk Outline

1

Algorithm for k = 1

2

Reducing k to 1

3

Putting it together

4

Continuous setting

Eric Price Sparse Fourier Transforms 22 / 36

slide-107
SLIDE 107

The Continuous Fourier Transform

Conversion between time and frequency domains

Time Domain Frequency Domain Fourier Transform The Fourier Transform x of an integrable function x : R → C is

  • x(f) =

+∞

−∞

x(t)e−2πiftdt

Eric Price Sparse Fourier Transforms 23 / 36

slide-108
SLIDE 108

The Continuous Fourier Transform

Conversion between time and frequency domains

Time Domain Frequency Domain Fourier Transform The Fourier Transform x of an integrable function x : R → C is

  • x(f) =

+∞

−∞

x(t)e−2πiftdt The Inverse Transform is: x(t) = +∞

−∞

  • x(f)e2πiftdf

Eric Price Sparse Fourier Transforms 23 / 36

slide-109
SLIDE 109

Why Continuous?

Eric Price Sparse Fourier Transforms 24 / 36

slide-110
SLIDE 110

Why Continuous?

Eric Price Sparse Fourier Transforms 24 / 36

slide-111
SLIDE 111

Why Continuous?

Eric Price Sparse Fourier Transforms 24 / 36

slide-112
SLIDE 112

Why Continuous?

Eric Price Sparse Fourier Transforms 24 / 36

slide-113
SLIDE 113

Thought Experiments

Frequency Time

Eric Price Sparse Fourier Transforms 25 / 36

slide-114
SLIDE 114

Thought Experiments

Frequency η Time T =

1 2η

Eric Price Sparse Fourier Transforms 25 / 36

slide-115
SLIDE 115

Thought Experiments

Frequency η Time T =

1 2η

Eric Price Sparse Fourier Transforms 25 / 36

slide-116
SLIDE 116

Thought Experiments

Frequency η Time T =

1 2η

Eric Price Sparse Fourier Transforms 25 / 36

slide-117
SLIDE 117

Thought Experiments

Frequency η Time T =

1 2η

inifinitely small η T > 1000 years

Eric Price Sparse Fourier Transforms 25 / 36

slide-118
SLIDE 118

Thought Experiments 2

x(t) = sink−1(ηt) Frequency η Time T =

1 2η

Eric Price Sparse Fourier Transforms 26 / 36

slide-119
SLIDE 119

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-120
SLIDE 120

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-121
SLIDE 121

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-122
SLIDE 122

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-123
SLIDE 123

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-124
SLIDE 124

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-125
SLIDE 125

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-126
SLIDE 126

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-127
SLIDE 127

What Guarantee Do We Want?

Frequency Time

  • x(f)

x(t) = e−2πift x ′(t) = x(t) · e−t/10000

  • x ′(f)

Discrete FT x ′ − x2 min

k-sparse xk

  • x −

xk2 For red signal : min

k-sparse xk

  • x −

xk2 = x2 DFT preserve the ℓ2 norm x ′ − x2 min

k-sparse xk

x − xk2

1 T

T

0 |x ′(t) − x(t)|2dt

min

k-sparse xk(t) 1 T

T

0 |x(t) − xk(t)|2dt

Eric Price Sparse Fourier Transforms 27 / 36

slide-128
SLIDE 128

Guarantee

Sample from x(t), which is approximated by a k-Fourier sparse xk(t) with η frequency separation. We recover an x ′(t) such that E

t∈[0,T]|x ′(t) − x(t)|2

E

t∈[0,T]|x(t) − xk(t)|2

As long as:

◮ T O( log2(FT)

η

)

◮ Time, # samples O(k log(FT) log2(k)). Eric Price Sparse Fourier Transforms 28 / 36

slide-129
SLIDE 129

Previous Works and Our Results

Algorithm Duration Robust Sample/Time BCGLS, 12 k·optimal poor sublinear Moitra, 15

  • ptimal

poly(k) linear Ours log2(k)·optimal O(1) sublinear

2 η log(k) η log2(k) η

Eric Price Sparse Fourier Transforms 29 / 36

slide-130
SLIDE 130

Previous Works and Our Results

Algorithm Duration Robust Sample/Time BCGLS, 12 k·optimal poor sublinear Moitra, 15

  • ptimal

poly(k) linear Ours log2(k)·optimal O(1) sublinear 2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 29 / 36

slide-131
SLIDE 131

Previous Works and Our Results

Algorithm Duration Robust Sample/Time BCGLS, 12 k·optimal poor sublinear Moitra, 15

  • ptimal

poly(k) linear Ours log2(k)·optimal O(1) sublinear 2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 29 / 36

slide-132
SLIDE 132

Previous Works and Our Results

Algorithm Duration Robust Sample/Time BCGLS, 12 k·optimal poor sublinear Moitra, 15

  • ptimal

poly(k) linear Ours log2(k)·optimal O(1) sublinear 2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 29 / 36

slide-133
SLIDE 133

Previous Works and Our Results

Algorithm Duration Robust Sample/Time BCGLS, 12 k·optimal poor sublinear Moitra, 15

  • ptimal

poly(k) linear Ours log2(k)·optimal O(1) sublinear 2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 29 / 36

slide-134
SLIDE 134

Main Results

Frequency Time

  • xk(f)

F xk(t) T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-135
SLIDE 135

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2

  • x(f)

F x(t) T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-136
SLIDE 136

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2

  • x ′(f)

F x ′(t) T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-137
SLIDE 137

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-138
SLIDE 138

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-139
SLIDE 139

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-140
SLIDE 140

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2 1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-141
SLIDE 141

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-142
SLIDE 142

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-143
SLIDE 143

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

Duration

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-144
SLIDE 144

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

Duration

T = log(k)

η

, T = log2(k)

η

O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-145
SLIDE 145

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

Duration

T = log(k)

η

, T = log2(k)

η

Samples/Time O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-146
SLIDE 146

Main Results

Frequency Time N2 := 1

T

T

0 |g(t)|2dt + δ k

  • i=1

|vi|2 F T

  • xk(f) ,

x ′(f) xk(t) ,x ′(t)

Tone Estimation

1 T k

  • i=1

T

0 | vie2πifit − v ′ i e2πif ′

i t |2dt N2

Frequency Estimation

|fi − f ′

i | 1 Tρ, ρ2 := SNR :=

  • i |vi|2

N2

Signal Estimation

1 T

T

0 | k

  • i=1

vie2πifit − v ′

i e2πif ′

i t |2dt N2

Duration

T = log(k)

η

, T = log2(k)

η

Samples/Time O(k log(FT) log2(k))

Eric Price Sparse Fourier Transforms 30 / 36

slide-147
SLIDE 147

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-148
SLIDE 148

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-149
SLIDE 149

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-150
SLIDE 150

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

1 T

T

0 |a1(t) - a′ 1(t)|2dt

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-151
SLIDE 151

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

1 T

T

0 |a1(t) - a′ 1(t)|2dt 1 T

T

0 |a2(t) - a′ 2(t)|2dt

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-152
SLIDE 152

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

1 T

T

0 |a1(t) - a′ 1(t)|2dt 1 T

T

0 |a2(t) - a′ 2(t)|2dt 1 T

T

0 |a3(t) - a′ 3(t)|2dt

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-153
SLIDE 153

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

1 T

T

0 |a1(t) - a′ 1(t)|2dt 1 T

T

0 |a2(t) - a′ 2(t)|2dt 1 T

T

0 |a3(t) - a′ 3(t)|2dt

+ + k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-154
SLIDE 154

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |ai(t) - a′ i (t)|2dt

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-155
SLIDE 155

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-156
SLIDE 156

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-157
SLIDE 157

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-158
SLIDE 158

Tone Estimation to Signal Estimation

Frequency

  • x(f)

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

k

i=1 1 T

T

0 |∆i(t)|2dt

N2

Tone Estimation

N2 1

T

T

0 | k i=1∆i(t)|2dt Signal Estimation

= 1

T

T

0 | xk(t) − x ′(t)|2dt

Eric Price Sparse Fourier Transforms 31 / 36

slide-159
SLIDE 159

Extremely Simplified Proof

Goal : k k

i=1 y2 i (k i=1 yi)2

(k

i=1 yi)2

= diagonal terms + off-diagonal terms = k

i=1 y2 i + i=j yiyj

k

i=1 y2 i + i=j 1 2(y2 i + y2 j )

= k k

i=1 y2 i

Eric Price Sparse Fourier Transforms 32 / 36

slide-160
SLIDE 160

Extremely Simplified Proof

Goal : k k

i=1 y2 i (k i=1 yi)2

(k

i=1 yi)2

= diagonal terms + off-diagonal terms = k

i=1 y2 i + i=j yiyj

k

i=1 y2 i + i=j 1 2(y2 i + y2 j )

= k k

i=1 y2 i

Eric Price Sparse Fourier Transforms 32 / 36

slide-161
SLIDE 161

Extremely Simplified Proof

Goal : k k

i=1 y2 i (k i=1 yi)2

(k

i=1 yi)2

= diagonal terms + off-diagonal terms = k

i=1 y2 i + i=j yiyj

k

i=1 y2 i + i=j 1 2(y2 i + y2 j )

= k k

i=1 y2 i

Eric Price Sparse Fourier Transforms 32 / 36

slide-162
SLIDE 162

Extremely Simplified Proof

Goal : k k

i=1 y2 i (k i=1 yi)2

(k

i=1 yi)2

= diagonal terms + off-diagonal terms = k

i=1 y2 i + i=j yiyj

k

i=1 y2 i + i=j 1 2(y2 i + y2 j )

= k k

i=1 y2 i

Eric Price Sparse Fourier Transforms 32 / 36

slide-163
SLIDE 163

Extremely Simplified Proof

Goal : k k

i=1 y2 i (k i=1 yi)2

(k

i=1 yi)2

= diagonal terms + off-diagonal terms = k

i=1 y2 i + i=j yiyj

k

i=1 y2 i + i=j 1 2(y2 i + y2 j )

= k k

i=1 y2 i

Eric Price Sparse Fourier Transforms 32 / 36

slide-164
SLIDE 164

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-165
SLIDE 165

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-166
SLIDE 166

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-167
SLIDE 167

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms = k

i=1 1 T

T

0 | ∆i(t) |2dt + i=j 1 T

T

0 ∆i(t)∆j(t) dt

T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-168
SLIDE 168

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms = k

i=1 1 T

T

0 | ∆i(t) |2dt + i=j 1 T

T

0 ∆i(t)∆j(t) dt

(1+log2(k)

Tη

) · k

i=1 1 T

T

0 | ∆i(t) |2dt

T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-169
SLIDE 169

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms = k

i=1 1 T

T

0 | ∆i(t) |2dt + i=j 1 T

T

0 ∆i(t)∆j(t) dt

  • k

i=1 1 T

T

0 | ∆i(t) |2dt for T > log2(k)/η

T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-170
SLIDE 170

Simplifed Proof

Define ∆i(t) = ai(t) − a′

i (t) = vie2πifit − v ′ i e2πif ′

i t

Goal : k

i=1 1 T

T

0 |∆i(t)|2dt 1 T

T

0 | k i=1 ∆i(t)|2dt 1 T

T

0 | k i=1∆i(t)|2dt

= diagonal terms + off-diagonal terms = k

i=1 1 T

T

0 | ∆i(t) |2dt + i=j 1 T

T

0 ∆i(t)∆j(t) dt

  • k

i=1 1 T

T

0 | ∆i(t) |2dt for T > log2(k)/η

T is large enough, ∆i(t) is more likely orthogonal to ∆j(t), ∀i = j

Eric Price Sparse Fourier Transforms 33 / 36

slide-171
SLIDE 171

Open questions

2 η log(k) η log2(k) η

Eric Price Sparse Fourier Transforms 34 / 36

slide-172
SLIDE 172

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 34 / 36

slide-173
SLIDE 173

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 34 / 36

slide-174
SLIDE 174

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 34 / 36

slide-175
SLIDE 175

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound

Eric Price Sparse Fourier Transforms 34 / 36

slide-176
SLIDE 176

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound Can we reconstruct a signal x ′(t) without recovering each (vi, fi) nicely?

Eric Price Sparse Fourier Transforms 34 / 36

slide-177
SLIDE 177

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound Can we reconstruct a signal x ′(t) without recovering each (vi, fi) nicely? Noise is exponentially small in k, how small duration T can we pick?

Eric Price Sparse Fourier Transforms 34 / 36

slide-178
SLIDE 178

Open questions

2Ω(k) kO(1) O(log k) O(1)

2 η log(k) η log2(k) η

Ours, Upper bound Moitra, 15 Upper bound Moitra, 15 Lower bound Can we reconstruct a signal x ′(t) without recovering each (vi, fi) nicely? Noise is exponentially small in k, how small duration T can we pick? Improve our constant approximation result to (1 ± ǫ) approximation by increasing the sample duration T?

Eric Price Sparse Fourier Transforms 34 / 36

slide-179
SLIDE 179

Summary

DFT setting: logd log n far from optimal in d dimensions. Continuous setting: more to learn.

Thank You

Eric Price Sparse Fourier Transforms 35 / 36

slide-180
SLIDE 180

Eric Price Sparse Fourier Transforms 36 / 36

slide-181
SLIDE 181

Eric Price Sparse Fourier Transforms 36 / 36