tutorial on sparse fourier transforms - mit csail · the fourier transform conversion between time...

Tutorial on Sparse Fourier Transforms

Eric Price

UT Austin

Eric Price Tutorial on Sparse Fourier Transforms 1 / 27

The Fourier TransformConversion between time and frequency domains

Time Domain Frequency Domain

Fourier Transform

Displacement of Air Concert A

The Fourier Transform is Ubiquitous

Audio Video Medical Imaging

Radar GPS Oil Exploration

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 mechanicalcalculations.

– Carl Friedrich Gauss, 1805

By hand: 22n log n seconds. [Danielson-Lanczos, 1942]Can we do better?

When can we compute the FourierTransform in sublinear time?

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

Fast Fourier Transform: O(n log n) time. [Cooley-Tukey, 1965]

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

By hand: 22n log n seconds. [Danielson-Lanczos, 1942]

Can we do better?

By hand: 22n log n seconds. [Danielson-Lanczos, 1942]Can we do much better?

Idea: Leverage SparsityOften the Fourier transform is dominated by a small number of peaks:

Time Signal Frequency(Exactly sparse)

Frequency(Approximately sparse)

Sparsity is common:

Audio Video MedicalImaging

Goal of this workshop: sparse Fourier transformsFaster Fourier Transform on sparse data.

Sparsity is common:

Classes of sparse Fourier transform algorithmsFor recovering a k -sparse signal in n dimensions.

Exact sparsity, deterministic algorithm

I Vandermonde matrix: 2k samples sufficientI Syndrome decoding for Reed-Solomon codingI Berlekamp-Massey: O(k2 + k(log log n)c) time.

Approximate sparsity, 2−k failure probability

I Compressed sensing, using Restricted Isometry PropertyI O(k log4 n) samples, O(n logc n) time.

Today: Approximate sparsity, 1/4 or 1/nc probability.

I Using hashingI O(k logc n) samples, O(k logc n) time.

Exact sparsity, deterministic algorithmI Vandermonde matrix: 2k samples sufficient

I Syndrome decoding for Reed-Solomon codingI Berlekamp-Massey: O(k2 + k(log log n)c) time.

Exact sparsity, deterministic algorithmI Vandermonde matrix: 2k samples sufficientI Syndrome decoding for Reed-Solomon coding

I Berlekamp-Massey: O(k2 + k(log log n)c) time.

Exact sparsity, deterministic algorithmI Vandermonde matrix: 2k samples sufficientI Syndrome decoding for Reed-Solomon codingI Berlekamp-Massey: O(k2 + k(log log n)c) time.

Approximate sparsity, 2−k failure probabilityI Compressed sensing, using Restricted Isometry Property

I O(k log4 n) samples, O(n logc n) time.Today: Approximate sparsity, 1/4 or 1/nc probability.

Approximate sparsity, 2−k failure probabilityI Compressed sensing, using Restricted Isometry PropertyI O(k log4 n) samples, O(n logc n) time.

Today: Approximate sparsity, 1/4 or 1/nc probability.I Using hashing

I O(k logc n) samples, O(k logc n) time.

Today: Approximate sparsity, 1/4 or 1/nc probability.I Using hashingI O(k logc n) samples, O(k logc n) time.

Kinds of Fourier transform

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

n∑j=1

ωijxj

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

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

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

Hadamard transform: x ∈ C2×2×···×2:

(−1)〈i,j〉xj

n∑j=1

ωijxj

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

(−1)〈i,j〉xj

n∑j=1

ωijxj

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

(−1)〈i,j〉xj

n∑j=1

ωijxj

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

(−1)〈i,j〉xj

n∑j=1

ωijxj

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

(−1)〈i,j〉xj

n∑j=1

ωijxj

xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2

(−1)〈i,j〉xj

Generic Algorithm Outline

Goal: given access to x , compute x ≈ xI Exact case: x is k -sparse, x = x (maybe to log n bits of precision)

I Approximate case:

‖x − x‖2 6 (1 + ε) mink -sparse xk

‖x − xk‖2

I With “good” probability.

1 Algorithm for k = 1 (exact or approximate)2 Method to reduce to k = 1 case

I Split x into O(k) “random” partsI Can sample time domain of the parts.

F O(k log k) time to get one sample from each of the k parts.

3 Finds “most” of signal; repeat on residual

Goal: given access to x , compute x ≈ xI Exact case: x is k -sparse, x = x (maybe to log n bits of precision)I Approximate case:

‖x − xk‖2

1 Algorithm for k = 1 (exact or approximate)

2 Method to reduce to k = 1 case

‖x − xk‖2

Permute Filters O(k)

1-sparse recovery

x x ′

‖x − xk‖2

1-sparse recovery

x x ′

‖x − xk‖2

I Split x into O(k) “random” parts

I Can sample time domain of the parts.

1-sparse recovery

x x ′

‖x − xk‖2

1-sparse recovery

x x ′

‖x − xk‖2

1-sparse recovery

x x ′

‖x − xk‖2

Talk Outline

1 Algorithm for k = 1

2 Reducing k to 1

3 Putting it together

Talk Outline

2 Reducing k to 1

Talk Outline

2 Reducing k to 1

Talk Outline

2 Reducing k to 1

Algorithm for k = 1: one dimension, exact case

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

{a if i = t0 otherwise

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.)

x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

x0 = a

x1 = aωt

x1/x0 = ωt =⇒ t .

x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

x0 = a x1 = aωt

x1/x0 = ωt =⇒ t . �

x0 = a x1 = aωt

x1/x0 = ωt =⇒ t . �

Algorithm for k = 1: one dimension, approximate caseLemmaSuppose x is approximately 1-sparse:

|xt |/‖x‖2 > 90%.

Then we can recover it with O(log n) samples and O(log2 n) time.

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. �

|xt |/‖x‖2 > 90%.

x1/x0 = ωt

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. �

|xt |/‖x‖2 > 90%.

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 .Error correcting code with efficient recovery =⇒ lemma. �

|xt |/‖x‖2 > 90%.

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 .

Error correcting code with efficient recovery =⇒ lemma. �

|xt |/‖x‖2 > 90%.

xc2/x0 = ωc2t + noise

|xt |/‖x‖2 > 90%.

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. �

Algorithm for k = 1: Hadamard settingLevin ’93, improving upon Goldreich-Levin ’89

xi =∑

(−1)〈i,j〉xj

LemmaSuppose x is approximately 1-sparse:

|xt |/‖x‖2 > 90%.

We have sign(xr ) = sign((−1)〈r ,t〉xt) with 9/10 probability over r .Therefore for any i , with 8/10 probability over r ,

sign(xi+r

xr) = sign((−1)〈i,t〉)

Choose i to be the O(log n) rows of generator matrix for constantrate and distance binary code.

xi =∑

(−1)〈i,j〉xj

|xt |/‖x‖2 > 90%.

sign(xi+r

xr) = sign((−1)〈i,t〉)

xi =∑

(−1)〈i,j〉xj

|xt |/‖x‖2 > 90%.

We have sign(xr ) = sign((−1)〈r ,t〉xt) with 9/10 probability over r .

Therefore for any i , with 8/10 probability over r ,

sign(xi+r

xr) = sign((−1)〈i,t〉)

xi =∑

(−1)〈i,j〉xj

|xt |/‖x‖2 > 90%.

sign(xi+r

xr) = sign((−1)〈i,t〉)

xi =∑

(−1)〈i,j〉xj

|xt |/‖x‖2 > 90%.

sign(xi+r

xr) = sign((−1)〈i,t〉)

Talk Outline

2 Reducing k to 1

Algorithm for general k

Reduce general k to k = 1.

“Filters”: partition frequencies intoO(k) buckets.

I Sample from time domain of eachbucket with O(log n) overhead.

I Recovered by k = 1 algorithm

Most frequencies alone in bucket.Random permutation

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 thatx − x ′ is k/2-sparse.

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

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

Most frequencies alone in bucket.

Random permutation

1-sparse recovery

x x ′

Random permutation

1-sparse recovery

x x ′

Random permutation

1-sparse recovery

x x ′

Random permutation

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

1-sparse recovery

x x ′

Going from finding most coordinates to finding allx

1-sparse recovery

Partial k -sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

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

Going from finding most coordinates to finding allx − x ′

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

1-sparse recovery

x x ′

Repeat, k → k/2→ k/4→ · · ·

How do filters work?x

Consider the√

n ×√

n 2d setting.

Get answer by FFT on rows, then FFT on resulting columns.What if I just take the FFT y r of a random row r?For any column z = x∗,c ∈ C

√n we have in the corresponding time

domainzr = y r

With O(√

n log n) time, get samples from time domains of all√

ncolumns.If column is 1-sparse, recover it with O(1) row FFTs

I For approximate sparsity, O(log n) row FFTs.

If k =√

n random nonzeros, expect to recover most of them.

Consider the√

n ×√

n 2d setting.Get answer by FFT on rows, then FFT on resulting columns.

What if I just take the FFT y r of a random row r?For any column z = x∗,c ∈ C

domainzr = y r

With O(√

If k =√

Consider the√

n ×√

n 2d setting.Get answer by FFT on rows, then FFT on resulting columns.

What if I just take the FFT y r of a random row r?For any column z = x∗,c ∈ C

domainzr = y r

With O(√

If k =√

Consider the√

n ×√

n 2d setting.Get answer by FFT on rows, then FFT on resulting columns.What if I just take the FFT y r of a random row r?

For any column z = x∗,c ∈ C√

n we have in the corresponding timedomain

zr = y rc

With O(√

If k =√

Consider the√

n ×√

n 2d setting.Get answer by FFT on rows, then FFT on resulting columns.What if I just take the FFT y r of a random row r?For any column z = x∗,c ∈ C

domainzr = y r

With O(√

If k =√

Consider the√

n ×√

domainzr = y r

With O(√

ncolumns.

If column is 1-sparse, recover it with O(1) row FFTs

If k =√

Consider the√

n ×√

domainzr = y r

With O(√

If k =√

Consider the√

n ×√

domainzr = y r

With O(√

If k =√

Consider the√

n ×√

domainzr = y r

With O(√

If k =√

Filters more generally

Fourier transform switches multiplication and convolution.

Choose a filter F so both F and F are sparse

I F is O(k)-sparse and F is (approximately) O(n/k) sparse.I Last slide: F is row, F is column

For various r , compute k -dimensional Fourier transform ofyi = xi+r Fi .

Gives the r th time domain sample of x · shift(F ) for k shifts of F .

Choose a filter F so both F and F are sparse

I F is O(k)-sparse and F is (approximately) O(n/k) sparse.I Last slide: F is row, F is column

Choose a filter F so both F and F are sparseI F is O(k)-sparse and F is (approximately) O(n/k) sparse.

I Last slide: F is row, F is column

Choose a filter F so both F and F are sparseI F is O(k)-sparse and F is (approximately) O(n/k) sparse.I Last slide: F is row, F is column

Hadamard setting: full algorithm

F = span(A) for any A ∈ Flog n×log B2

For any r ∈ span(A)⊥, compute Hadamard transform of

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

Gives O(k log(n/k)) total samples and O(k log(n/k) log k) time

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.

If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.

Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.

Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.

For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.

Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same coset withprobability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) time

Repeat with k → k/2→ k/4→ . . .

yi = xAi+r

Gives r th time domain sample of x restricted to all B cosets of A⊥.If A is chosen randomly, then any two i , j land in same cosetwith probability 1/B.Each coordinate is alone with probability 1 − k/B.Take log(n/k) different r to solve the 1-sparse problem on coset.For B = O(k), expect to recover “most” coordinates.Takes O(k log(n/k)) samples and O(k log(n/k) log k) timeRepeat with k → k/2→ k/4→ . . .

Difficulty in other settings

Not enough filters F that are “perfect” (F and F are indicators)

Two dimensions:

I Can look at the columns or the rowsI Some inputs will cause collisions for any projection.

Works if you assume coordinates randomly distributed [GHIKPS,Pawar-Ramchandran]

I Peeling procedureI Still doesn’t work for 1 dimension, n = 2`.

For worst-case inputs, need other filters

Not enough filters F that are “perfect” (F and F are indicators)Two dimensions:

I Can look at the columns or the rows

I Some inputs will cause collisions for any projection.Works if you assume coordinates randomly distributed [GHIKPS,Pawar-Ramchandran]

I Peeling procedure

I Still doesn’t work for 1 dimension, n = 2`.

A different style of filterGMS05, HIKP12, IKP14, IK14

Filter (time): k uniformly spaced Filter (frequency): n/k uniformly spacedFilter (time): Gaussian · sinc Filter (frequency): Gaussian * rectangle

Previous slides used comb filter

Instead, make filter so F is large on an interval.We can permute the frequencies:

x ′i = xσi =⇒ xi = xσ−1i

This changes the coordinates in an interval (unlike in a comb).Allows us to convert worst case to random case.

Previous slides used comb filterInstead, make filter so F is large on an interval.

We can permute the frequencies:

Previous slides used comb filterInstead, make filter so F is large on an interval.We can permute the frequencies:

This changes the coordinates in an interval (unlike in a comb).

Allows us to convert worst case to random case.

Talk Outline

2 Reducing k to 1

How can you hope for sublinear time?Time Frequency

n-dimensional DFT:O(n log n)x → x

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

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

Algorithm for exactly sparse signalsOriginal signal x Goal x

LemmaIf t is isolated in its bucket and in the “super-pass” region, the value bwe compute for its bucket satisfies

b = xt .

Computing the b for all O(k) buckets takes O(k log n) time.

Algorithm for exactly sparse signalsComputed F ·x Filtered signal F ∗x

b = xt .

Algorithm for exactly sparse signalsF ·x aliased to k terms Filtered signal F ∗x

b = xt .

Algorithm for exactly sparse signalsF ·x aliased to k terms Computed samples of F ∗x

b = xt .

Algorithm for exactly sparse signalsF ·x aliased to k terms Computed samples of F ∗x

b = xt .

Algorithm for exactly sparse signalsF ·x aliased to k terms Knowledge about x

b = xt .

AlgorithmLemmaFor most t, the value b we compute for its bucket satisfies

b = xt .

Time-shift x by one and repeat: b ′ = xtωt .

Divide to get b ′/b = ωt

=⇒ can compute t .I Just like our 1-sparse recovery algorithm, x1/x0 = ωt .

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

1-sparse recovery

x x ′

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

b = xt .

Divide to get b ′/b = ωt

=⇒ can compute t .I Just like our 1-sparse recovery algorithm, x1/x0 = ωt .

1-sparse recovery

x x ′

b = xt .

Divide to get b ′/b = ωt =⇒ can compute t .

I Just like our 1-sparse recovery algorithm, x1/x0 = ωt .Gives partial sparse recovery: x ′ such that x − x ′ is k/2-sparse.

1-sparse recovery

x x ′

b = xt .

Divide to get b ′/b = ωt =⇒ can compute t .I Just like our 1-sparse recovery algorithm, x1/x0 = ωt .

1-sparse recovery

x x ′

b = xt .

1-sparse recovery

x x ′

b = xt .

1-sparse recovery

x x ′

Repeat k → k/2→ k/4→ · · ·

O(k log n) time sparse Fourier transform. �

b = xt .

1-sparse recovery

x x ′

State of the ArtAlgorithms based on two kinds of filters:

I Comb filter works for

F Hadamard transform in the worst caseF > 1 dimensional transform, or n = (pq)`, in the average case.

I Interval filter works for

F Constant dimensional transform in the worst case, n has Θ(k)-sizedfactors.

Exactly sparse: “optimal” is O(k) samples and O(k log k) time(and log(n/k) factor larger for Hadamard)

I Comb filter: optimal when it worksI Interval filter: O(k log n) samples and time

Approximately sparse: “optimal” is O(k log(n/k)) samples andO(k log(n/k) log n) time

I Comb filter: optimal when it worksI Interval filter: optimal samples OR optimal time OR

logc log n-competitive mixture.

Thank You

I Comb filter works for

F Hadamard transform in the worst caseF > 1 dimensional transform, or n = (pq)`, in the average case.

Thank You

I Comb filter works forF Hadamard transform in the worst case

F > 1 dimensional transform, or n = (pq)`, in the average case.I Interval filter works for

Thank You

I Comb filter works forF Hadamard transform in the worst caseF > 1 dimensional transform, or n = (pq)`, in the average case.

Thank You

I Interval filter works forF Constant dimensional transform in the worst case, n has Θ(k)-sized

factors.

Thank You

factors.Exactly sparse: “optimal” is O(k) samples and O(k log k) time(and log(n/k) factor larger for Hadamard)

Thank You

I Comb filter: optimal when it works

I Interval filter: O(k log n) samples and timeApproximately sparse: “optimal” is O(k log(n/k)) samples andO(k log(n/k) log n) time

Thank You

I Comb filter: optimal when it works

I Interval filter: optimal samples OR optimal time ORlogc log n-competitive mixture.

Thank You

Thank YouEric Price Tutorial on Sparse Fourier Transforms 26 / 27

tutorial on sparse fourier transforms - mit csail · the fourier transform conversion between time...

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