Tutorial on Sparse Fourier Transforms - MIT CSAIL · The Fourier Transform Conversion between time and frequency domains Time Domain Frequency Domain Fourier Transform Displacement

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?







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:








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

xi =

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:

xi =

n∑j

(−1)〈i,j〉xj




xi =

n∑j=1

ωijxj


xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2



xi =

n∑j

(−1)〈i,j〉xj




xi =

n∑j=1

ωijxj


xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2



xi =

n∑j

(−1)〈i,j〉xj




xi =

n∑j=1

ωijxj


xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2



xi =

n∑j

(−1)〈i,j〉xj




xi =

n∑j=1

ωijxj


xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2



xi =

n∑j

(−1)〈i,j〉xj




xi =

n∑j=1

ωijxj


xi1,i2 =

n1∑j1=1

n2∑j2=1

ωi1j11 ω

i2j22 xj1,j2



xi =

n∑j

(−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










‖x − xk‖2










‖x − xk‖2


1 Algorithm for k = 1 (exact or approximate)

2 Method to reduce to k = 1 case








‖x − xk‖2










‖x − xk‖2








Permute Filters O(k)

1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′



‖x − xk‖2









1-sparse recovery

1-sparse recovery

1-sparse recovery

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

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′



‖x − xk‖2









1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′



‖x − xk‖2









1-sparse recovery

1-sparse recovery

1-sparse recovery

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

x :

t

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

xi =

{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.)



x :

t


xi =



x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

�




x :

t


xi =



x0 = a

x1 = aωt

x1/x0 = ωt =⇒ t .

�




x :

t


xi =



x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

�




x :

t


xi =



x0 = a x1 = aωt

x1/x0 = ωt =⇒ t .

�




x :

t


xi =



x0 = a x1 = aωt

x1/x0 = ωt =⇒ t . �




x :

t


xi =



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%.







|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 =∑

j

(−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 =∑

j

(−1)〈i,j〉xj


|xt |/‖x‖2 > 90%.



sign(xi+r

xr) = sign((−1)〈i,t〉)




xi =∑

j

(−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 =∑

j

(−1)〈i,j〉xj


|xt |/‖x‖2 > 90%.



sign(xi+r

xr) = sign((−1)〈i,t〉)




xi =∑

j

(−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

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 thatx − x ′ is k/2-sparse.



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





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′








Most frequencies alone in bucket.

Random permutation


1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′









Random permutation


1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′









Random permutation


1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′









Random permutation


1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′










1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′




Going from finding most coordinates to finding allx


1-sparse recovery

1-sparse recovery

1-sparse recovery

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

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery


x x ′


Repeat, k → k/2→ k/4→ · · ·





1-sparse recovery

1-sparse recovery

1-sparse recovery

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

c

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

c

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

c

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

c

With O(√




If k =√




Consider the√

n ×√



domainzr = y r

c

With O(√


ncolumns.

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


If k =√




Consider the√

n ×√



domainzr = y r

c

With O(√




If k =√




Consider the√

n ×√



domainzr = y r

c

With O(√




If k =√




Consider the√

n ×√



domainzr = y r

c

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







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







yi = xAi+r







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 .





b = xt .





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

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. �



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

1-sparse recovery

1-sparse recovery

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

1-sparse recovery

1-sparse recovery

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

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′




b = xt .






1-sparse recovery

1-sparse recovery

1-sparse recovery

1-sparse recovery

x x ′




b = xt .






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. �



b = xt .






1-sparse recovery

1-sparse recovery

1-sparse recovery

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











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










Thank You










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 You










Thank YouEric Price Tutorial on Sparse Fourier Transforms 26 / 27



Documents

Tutorial on Sparse Fourier Transforms - MIT CSAIL · The Fourier Transform Conversion between time and frequency domains Time Domain Frequency Domain Fourier Transform Displacement