# Алгоритмы быстрого вычисления разреженного преобразования Фурье (Sparse FFT), осень 2015: Сигналы с разреженным спектром Фурье: определения и баз

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• Sparse Fourier Transform(lecture 1)

Michael Kapralov1

1IBM Watson EPFL

St. Petersburg CS ClubNovember 2015

1 / 73

• Given x Cn, compute the Discrete Fourier Transform (DFT) ofx :

xi =1n

j[n]

xj ij ,

where = e2i/n is the n-th root of unity.

Assume that n is a power of 2.

compression schemes(JPEG, MPEG)

signal processingdata analysis

imaging (MRI, NMR)

2 / 73

• Given x Cn, compute the Discrete Fourier Transform (DFT) ofx :

xi =1n

j[n]

xj ij ,

where = e2i/n is the n-th root of unity.

Assume that n is a power of 2.

compression schemes(JPEG, MPEG)

signal processingdata analysis

imaging (MRI, NMR)

2 / 73

• Given x Cn, compute the Discrete Fourier Transform (DFT) ofx :

xi =1n

j[n]

xj ij ,

where = e2i/n is the n-th root of unity.

Assume that n is a power of 2.

compression schemes(JPEG, MPEG)

signal processingdata analysis

imaging (MRI, NMR)

2 / 73

• DFT has numerous applications:

3 / 73

• Fast Fourier Transform (FFT)

Computes Discrete Fourier Transform (DFT) of a length nsignal in O(n logn) time

Cooley and Tukey, 1964

Gauss, 1805

Code=FFTW (Fastest Fourier Transform in the West)

4 / 73

• Fast Fourier Transform (FFT)

Computes Discrete Fourier Transform (DFT) of a length nsignal in O(n logn) time

Cooley and Tukey, 1964

Gauss, 1805

Code=FFTW (Fastest Fourier Transform in the West)

4 / 73

• Fast Fourier Transform (FFT)

Computes Discrete Fourier Transform (DFT) of a length nsignal in O(n logn) time

Cooley and Tukey, 1964

Gauss, 1805

Code=FFTW (Fastest Fourier Transform in the West)

5 / 73

• Fast Fourier Transform (FFT)

Computes Discrete Fourier Transform (DFT) of a length nsignal in O(n logn) time

Cooley and Tukey, 1964

Gauss, 1805

Code=FFTW (Fastest Fourier Transform in the West)

6 / 73

• Sparse FFT

Say that x is k -sparse if x has k nonzero entries

Say that x is approximately k -sparse if x is close to k -sparse insome norm (`2 for this lecture)

-0.0004

-0.0002

0

0.0002

0.0004

-1000 -500 0 500 1000

time

0

0.2

0.4

0.6

0.8

1

-1000 -500 0 500 1000

frequency

7 / 73

• Sparse FFT

Say that x is k -sparse if x has k nonzero entries

Say that x is approximately k -sparse if x is close to k -sparse insome norm (`2 for this lecture)

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

-1000 -500 0 500 1000

time

0

0.2

0.4

0.6

0.8

1

-1000 -500 0 500 1000

frequency

8 / 73

• Sparse approximations

JPEG=

Given x , compute x , then keep top k coefficients only for k N

Used in image and video compression schemes(e.g. JPEG, MPEG)

9 / 73

• Sparse approximations

JPEG=

Given x , compute x , then keep top k coefficients only for k N

Used in image and video compression schemes(e.g. JPEG, MPEG)

10 / 73

• Computing approximation fast

Basic approach:

FFT computes x from x in O(n logn) time

compute top k coefficients in O(n) time.

Sparse FFT: directly computes k largest coefficients of x (approximately

formal def later)

Running time O(k log2 n) or faster

Sublinear time!

11 / 73

• Computing approximation fast

Basic approach:

FFT computes x from x in O(n logn) time

compute top k coefficients in O(n) time.

Sparse FFT: directly computes k largest coefficients of x (approximately

formal def later)

Running time O(k log2 n) or faster

Sublinear time!

11 / 73

• Sample complexity

Sample complexity=number of samples accessed in timedomain.

12 / 73

• Sample complexity

In medical imaging (MRI, NMR), one measures Fouriercoefficients x of imaged object x (which is often sparse)

13 / 73

• Sample complexity

In medical imaging (MRI, NMR), one measures Fouriercoefficients x of imaged object x (which is often sparse)

13 / 73

• Sample complexity

Measure x Cn, compute the Inverse Discrete FourierTransform (IDFT) of x :

xi =

j[n]xj ij .

Given x Cn, compute the Discrete Fourier Transform (DFT) ofx :

xi =1n

j[n]

xj ij .

Sample complexity=number of samples accessed in timedomain.

Governs the measurement complexity of imaging process.

14 / 73

• Sample complexity

Measure x Cn, compute the Inverse Discrete FourierTransform (IDFT) of x :

xi =

j[n]xj ij .

Given x Cn, compute the Discrete Fourier Transform (DFT) ofx :

xi =1n

j[n]

xj ij .

Sample complexity=number of samples accessed in timedomain.

Governs the measurement complexity of imaging process.

14 / 73

• Sample complexity

Measure x Cn, compute the Inverse Discrete FourierTransform (IDFT) of x :

xi =

j[n]xj ij .

Given x Cn, compute the Discrete Fourier Transform (DFT)of x:

xi =1n

j[n]

xj ij .

Sample complexity=number of samples accessed in timedomain.

Governs the measurement complexity of imaging process.

15 / 73

• Sample complexity

Measure x Cn, compute the Inverse Discrete FourierTransform (IDFT) of x :

xi =

j[n]xj ij .

Given x Cn, compute the Discrete Fourier Transform (DFT)of x:

xi =1n

j[n]

xj ij .

Sample complexity=number of samples accessed in timedomain.

Governs the measurement complexity of imaging process.

15 / 73

• Given access to signal x in time domain, find best k -sparseapproximation to x approximately

Minimize

1. runtime

2. number of samples

16 / 73

• Algorithms

Randomization Approximation Hashing Sketching . . .

Signal processing

Fourier transform Hadamard transform Filters Compressive sensing . . .

17 / 73

• Lecture 1: summary of techniques fromGilbert-Guha-Indyk-Muthukrishnan-Strauss02, Akavia-Goldwasser-Safra03,

Gilbert-Muthukrishnan-Strauss05, Iwen10, Akavia10,

Hassanieh-Indyk-Katabi-Price12a, Hassanieh-Indyk-Katabi-Price12b

Lecture 2: Algorithm with O(k logn) runtime (noiselesscase) Hassanieh-Indyk-Katabi-Price12b

Lecture 3: Algorithm with O(k log2 n) runtime (noisy case)Hassanieh-Indyk-Katabi-Price12b

Lecture 4: Algorithm with O(k logn) sample complexityIndyk-Kapralov-Price14, Indyk-Kapralov14

18 / 73

• Outline

1. Computing Fourier transform of 1-sparse signals fast

2. Sparsity k > 1: main ideas and challenges

19 / 73

• Outline

1. Computing Fourier transform of 1-sparse signals fast

2. Sparsity k > 1: main ideas and challenges

20 / 73

• Sparse Fourier Transform (k = 1)Warmup: x is exactly 1-sparse: xf = 0 when f 6= f for some f

-0.0004

-0.0002

0

0.0002

0.0004

-1000 -500 0 500 1000

time

0

0.2

0.4

0.6

0.8

1

-1000 -500 0 500 1000

frequency

f

Note: signal is a pure frequency

Need: find f and xf

21 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

f

unit circle

x0 = ax1 = a f

Pro: constant time algorithmCon: depends heavily on the signal being pure

22 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

f

unit circle

x0 = ax1 = a f

Pro: constant time algorithmCon: depends heavily on the signal being pure

22 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

f

unit circle

x0 = ax1 = a f

Pro: constant time algorithmCon: depends heavily on the signal being pure

23 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

f

unit circle

x0 = ax1 = a f

Pro: constant time algorithmCon: depends heavily on the signal being pure

24 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

f

unit circle

x0 = ax1 = a f

Pro: constant time algorithmCon: depends heavily on the signal being pure

25 / 73

• Two-point samplingInput signal x is a pure frequency, so xj = a f j

+noise

Sample x0,x1

We have

x0 = a

+noise

x1 = a f

+noise

So

x1/x0 =f

+noise

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