Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables

Abstract

This paper considers fast and provably accurate algorithms for approximating smooth functions on the d-dimensional torus, $f: \mathbb {T}^d \rightarrow \mathbb {C}$, that are sparse (or compressible) in the multidimensional Fourier basis. In particular, suppose that the Fourier series coefficients of f, $\{c_\mathbf{k} (f) \}_{\mathbf{k} \in \mathbb {Z}^d}$, are concentrated in a given arbitrary finite set $\mathscr {I} \subset \mathbb {Z}^d$ so that

$$\begin{aligned} \min _{\text{\O}mega \subset \mathscr {I} ~s.t.~ \left| \text{\O}mega \right| =s }\left\| f - \sum _{\mathbf{k} \in \text{\O}mega } c_\mathbf{k} (f) \, \mathbb {e}^{ -2 \pi \mathbb {i} {{\mathbf {k}}}\cdot \circ } \right\| _2 < \epsilon \Vert f \Vert _2 \end{aligned}$$

holds for $s \ll \left| \mathscr {I} \right|$ and $\epsilon \in (0,1)$ small. In such cases we aim to both identify a near-minimizing subset $\text{\O}mega \subset \mathscr {I}$ and accurately approximate its associated Fourier coefficients $\{ c_\mathbf{k} (f) \}_{\mathbf{k} \in \text{\O}mega }$ as rapidly as possible. In this paper we present both deterministic and explicit as well as randomized algorithms for solving this problem using $\mathscr {O}(s^2 d \log ^c (|\mathscr {I}|))$-time/memory and $\mathscr {O}(s d \log ^c (|\mathscr {I}|))$-time/memory, respectively. Most crucially, all of the methods proposed herein achieve these runtimes while simultaneously satisfying theoretical best s-term approximation guarantees which guarantee their numerical accuracy and robustness to noise for general functions. These results are achieved by modifying several different one-dimensional Sparse Fourier Transform (SFT) methods to subsample a function along a reconstructing rank-1 lattice for the given frequency set $\mathscr {I} \subset \mathbb {Z}^d$ in order to rapidly identify a near-minimizing subset $\text{\O}mega \subset \mathscr {I}$ as above without having to use anything about the lattice beyond its generating vector. This requires the development of new fast and low-memory frequency identification techniques capable of rapidly recovering vector-valued frequencies in $\mathbb {Z}^d$ as opposed to recovering simple integer frequencies as required in the univariate setting. Two different multivariate frequency identification strategies are proposed, analyzed, and shown to lead to their own best s-term approximation methods herein, each with different accuracy versus computational speed and memory tradeoffs.

Appendices

Proof of Theorem 1

We begin with a lemma which will be used to slightly improve the error bounds in Theorems 1 and 2 from their original statements.

Lemma 4

For ${{\mathbf {x}}}\in {\mathbb {C}}^K$ and $\mathscr {S}_\tau := \{ k \in [K] \mid |x_k| \ge \tau \}$, if $\tau \ge \frac{\Vert {{\mathbf {x}}}- {{\mathbf {x}}}_s^\mathrm {opt}\Vert _1}{s}$, then $| \mathscr {S}_\tau | \le 2s$ and

$$\begin{aligned} \Vert {{\mathbf {x}}}- {{\mathbf {x}}}|_{ \mathscr {S}_\tau }\Vert _{ 2 } \le \Vert {{\mathbf {x}}}- {{\mathbf {x}}}_{ 2s }^\mathrm {opt} \Vert _2 + \tau \sqrt{ 2s } . \end{aligned}$$

Proof

Ordering the entries of ${{\mathbf {x}}}$ in descending order (with ties broken arbitrarily) as $|x_{ k_1 }| \ge |x_{ k_2 }| \ge \ldots$, we first note that

$$\begin{aligned} \Vert {{\mathbf {x}}}- {{\mathbf {x}}}_s^\mathrm {opt}\Vert _1 \ge \sum _{ j = s + 1 }^{ 2s } |x_{ k_j }| \ge s |x_{ k_{ 2s } }|. \end{aligned}$$

By assumption then, $\tau \ge | x_{ k_{ 2s } }|$, and since $\mathscr {S}_\tau$ contains the $| \mathscr {S}_\tau |$-many largest entries of ${{\mathbf {x}}}$, we must have $\mathscr {S}_\tau \subset {{\,\mathrm{supp}\,}}({{\mathbf {x}}}_{ 2s }^\mathrm {opt})$. Note then that $| \mathscr {S}_\tau | \le 2s$. Finally, we calculate

$$\begin{aligned} \Vert {{\mathbf {x}}}- {{\mathbf {x}}}|_{ \mathscr {S}_\tau }\Vert _2&\le \Vert {{\mathbf {x}}}- {{\mathbf {x}}}_{ 2s }^\mathrm {opt} \Vert _2 + \Vert {{\mathbf {x}}}_{ 2s }^\mathrm {opt} - {{\mathbf {x}}}_{ \mathscr {S}_\tau }\Vert _2 \\&\le \Vert {{\mathbf {x}}}- {{\mathbf {x}}}_{ 2s }^\mathrm {opt} \Vert _2 + \sqrt{ \sum _{ k \in {{\,\mathrm{supp}\,}}({{\mathbf {x}}}_{ 2s }^\mathrm {opt}) \setminus \mathscr {S}_\tau } x_k^2 } \\&\le \Vert {{\mathbf {x}}}- {{\mathbf {x}}}_{ 2s }^\mathrm {opt} \Vert _2 + \tau \sqrt{ 2s } \end{aligned}$$

completing the proof. $\square$

We now give the proof of Theorem 1. For convenience we restate the theorem.

Theorem 1

For a signal $a \in \mathscr {W}(\mathbb { T }) \cap C( \mathbb { T } )$ corrupted by some arbitrary noise $\mu : \mathbb { T } \rightarrow {\mathbb {C}}$, Algorithm 3 of [16], denoted $\mathscr {A}_{ 2s, M }^{ \mathrm {sub} }$, will output a 2s-sparse coefficient vector ${{\mathbf {v}}}\in {\mathbb {C}}^M$ which

• 1. reconstructs every frequency of ${\hat{{{\mathbf {a}}}}} \in {\mathbb {C}}^M$, $\omega \in \mathscr {B}_M$, with corresponding Fourier coefficients meeting the tolerance

  $$\begin{aligned} | {\hat{a}}_\omega | > (4 + 2 \sqrt{ 2 }) \left( \frac{\Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_{ s }^\mathrm {opt}\Vert _1}{s} + \Vert {\hat{a}} - {\hat{{{\mathbf {a}}}}} \Vert _1 + \Vert \mu \Vert _\infty \right) , \end{aligned}$$

• 2. satisfies the $\ell ^\infty$ error estimate for recovered coefficients

  $$\begin{aligned} \Vert ({\hat{{{\mathbf {a}}}}} - {{\mathbf {v}}})|_{ {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) } \Vert _\infty \le \sqrt{ 2 } \left( \frac{\Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_{ s }^\mathrm {opt}\Vert _1}{s} + \Vert {\hat{a}} - {\hat{{{\mathbf {a}}}}} \Vert _1 + \Vert \mu \Vert _\infty \right) , \end{aligned}$$

• 3. satisfies the $\ell ^2$ error estimate

  $$\begin{aligned} \Vert {\hat{{{\mathbf {a}}}}} - {{\mathbf {v}}}\Vert _2&\le \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_{ 2s }^\mathrm {opt} \Vert _2 + \frac{(8 \sqrt{ 2 } + 6)\Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_s^\mathrm {opt} \Vert _1}{\sqrt{ s }} \\&\qquad + (8 \sqrt{ 2 } + 6)\sqrt{ s } (\Vert {\hat{a}} - {\hat{{{\mathbf {a}}}}} \Vert _1 + \Vert \mu \Vert _\infty ), \end{aligned}$$

• 4. and the number of required samples of a and the operation count for $\mathscr {A}_{ 2s, M }^{ \mathrm {sub} }$ are

  $$\begin{aligned} \mathscr {O} \left( \frac{s^2\log ^4 M}{\log s} \right) . \end{aligned}$$

The Monte Carlo variant of $\mathscr {A}_{ 2s, M }^\mathrm {sub}$, denoted $\mathscr {A}_{ 2s, M }^\mathrm {sub, MC}$, referred to by Corollary 4 of [16] satisfies all of the conditions (1)–(3) simultaneously with probability $(1 - \sigma ) \in [2/3, 1)$ and has number of required samples and operation count

$$\begin{aligned} \mathscr {O} \left( s \log ^3(M) \log \left( \frac{M}{\sigma } \right) \right) . \end{aligned}$$

The samples required by $\mathscr {A}_{ 2s, M }^\mathrm {sub, MC}$ are a subset of those required by $\mathscr {A}_{ 2s, M }^\mathrm {sub}$.

Proof

We refer to [16, Theorem 7] and its modification for noise robustness in [27, Lemma 4] for the proofs of properties (2) and (4). As for (1), [16, Lemma 6] and its modification in [27, Lemma 4] imply that any $\omega \in \mathscr {B}_M$ with $|{\hat{a}}_\omega | > 4 (\Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_s^\mathrm {opt}\Vert _1 / s + \Vert {\hat{a}} - {\hat{{{\mathbf {a}}}}} \Vert _1 + \Vert \mu \Vert _\infty ) =: 4\delta$ will be identified in [16, Algorithm 3]. An approximate Fourier coefficient for these and any other recovered frequencies is stored in the vector ${{\mathbf {x}}}$ which satisfies the same estimate in property (2) by the proof of [16, Theorem 7] and [27, Lemma 4]. However, only the 2s largest magnitude values of ${{\mathbf {x}}}$ will be returned in ${{\mathbf {v}}}$. We therefore analyze what happens when some of the potentially large Fourier coefficients corresponding to frequencies in $\mathscr {S}_{ 4\delta }$ do not have their approximations assigned to ${{\mathbf {v}}}$.

For the definition of $\mathscr {S}_\tau$ in Lemma 4 applied to ${\hat{{{\mathbf {a}}}}}$, we must have $| \mathscr {S}_{ 4\delta }| \le 2s = |{{\,\mathrm{supp}\,}}({{\mathbf {v}}})|$. If $\omega \in \mathscr {S}_{ 4\delta } \setminus {{\,\mathrm{supp}\,}}({{\mathbf {v}}})$, there must then exist some other $\omega ' \in {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) \setminus \mathscr {S}_{ 4\delta }$ which was identified and took the place of $\omega$ in ${{\,\mathrm{supp}\,}}({{\mathbf {v}}})$. For this to happen, $|{\hat{a}}_{ \omega ' }| \le 4\delta$ and $|x_{ \omega ' }| \ge |x_{ \omega }|$. But by property (2) (extended to all coefficients in ${{\mathbf {x}}}$), we know

$$\begin{aligned} 4 \delta + \sqrt{ 2 } \delta \ge |{\hat{a}}_{ \omega ' }| + \sqrt{ 2 } \delta \ge |x_{ \omega ' }| \ge |x_\omega | \ge |{\hat{a}}_\omega | - \sqrt{ 2 }\delta . \end{aligned}$$

Thus, any frequency in $\mathscr {S}_{ 4\delta }$ not chosen satisfies $|{\hat{a}}_\omega | \le (4 + 2\sqrt{ 2 }) \delta$, and so every frequency in $\mathscr {S}_{ (4 + 2\sqrt{ 2 }) \delta }$ is in fact identified in ${{\mathbf {v}}}$ verifying property (1).

As for property (3), we estimate the $\ell ^2$ error using property (2), Lemma 4, and the above argument as

$$\begin{aligned} \Vert {\hat{{{\mathbf {a}}}}} - {{\mathbf {v}}}\Vert _2&\le \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}|_{ {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) } \Vert _2 + \Vert ({\hat{{{\mathbf {a}}}}} - {{\mathbf {v}}})|_{ {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) } \Vert _2 \\&\le \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}|_{ \mathscr {S}_{ 4\delta } \cap {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) } \Vert _2 + \sqrt{ 2 } \delta \sqrt{ 2 s }\\&\le \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}|_{ \mathscr {S}_{ 4\delta } } \Vert _2 + \Vert {\hat{{{\mathbf {a}}}}}|_{ \mathscr {S}_{ 4\delta } \setminus {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) }\Vert _2 + 2 \delta \sqrt{ s } \\&\le \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_{ 2s }^\mathrm {opt} \Vert _2 + 4 \delta \sqrt{ 2s } + (4 + 2\sqrt{ 2 }) \delta \sqrt{ 2s } + 2 \delta \sqrt{ s }\\&= \Vert {\hat{{{\mathbf {a}}}}} - {\hat{{{\mathbf {a}}}}}_{ 2s }^\mathrm {opt} \Vert _2 + (8 \sqrt{ 2 } + 6) \sqrt{ s } \delta \end{aligned}$$

as desired. $\square$

Proof of Theorem 2

We again restate the theorem for convenience.

Theorem 2

For a signal $a \in \mathscr {W}(\mathbb { T }) \cap C( \mathbb { T } )$ corrupted by some arbitrary noise $\mu : \mathbb { T } \rightarrow {\mathbb {C}}$, and $1 \le r \le \frac{M}{36}$ Algorithm 1 of [27], denoted $\mathscr {A}_{ 2s, M }^{ \mathrm {disc} }$, will output a 2s-sparse coefficient vector ${{\mathbf {v}}}\in {\mathbb {C}}^M$ which

• 1. reconstructs every frequency of ${{\mathbf {F}}}_M \, {{\mathbf {a}}}\in {\mathbb {C}}^M$, $\omega \in \mathscr {B}_M$, with corresponding aliased Fourier coefficient meeting the tolerance

  $$\begin{aligned} | ({{\mathbf {F}}}_M \, {{\mathbf {a}}})_\omega | > 12(1 + \sqrt{ 2 }) \left( \frac{\Vert {{\mathbf {F}}}_M \, {{\mathbf {a}}}- ({{\mathbf {F}}}_M \, {{\mathbf {a}}})_s^\mathrm {opt}\Vert _1}{2s} + 2( \Vert {{\mathbf {a}}}\Vert _\infty M^{ -r } + \Vert \varvec{\mu } \Vert _{ \infty } )\right) , \end{aligned}$$

• 2. satisfies the $\ell ^\infty$ error estimate for recovered coefficients

  $$\begin{aligned} \Vert ({{\mathbf {F}}}_M \, {{\mathbf {a}}}- {{\mathbf {v}}})|_{ {{\,\mathrm{supp}\,}}({{\mathbf {v}}}) } \Vert _\infty \le 3 \sqrt{ 2 } \left( \frac{\Vert {{\mathbf {F}}}_M \, {{\mathbf {a}}}- ({{\mathbf {F}}}_M \, {{\mathbf {a}}})_s^\mathrm { opt }\Vert _1}{2s} + 2 (\Vert {{\mathbf {a}}}\Vert _\infty M^{ -r } + \Vert \varvec{\mu } \Vert _\infty ) \right) , \end{aligned}$$

• 3. satisfies the $\ell ^2$ error estimate

  $$\begin{aligned} \Vert {{\mathbf {F}}}_M \, {{\mathbf {a}}}- {{\mathbf {v}}}\Vert _2&\le \Vert {{\mathbf {F}}}_M \, {{\mathbf {a}}}- ({{\mathbf {F}}}_M \, {{\mathbf {a}}})_{ 2s }^\mathrm {opt} \Vert _2 + 38 \frac{\Vert {{\mathbf {F}}}_M \, {{\mathbf {a}}}- ({{\mathbf {F}}}_M \, {{\mathbf {a}}})_s^\mathrm {opt} \Vert _1}{\sqrt{ s }} \\&\quad + 152 \sqrt{ s } (\Vert {{\mathbf {a}}}\Vert _\infty M^{ -r } + \Vert \varvec{\mu } \Vert _\infty ), \end{aligned}$$

• 4. and the number of required samples of ${{\mathbf {a}}}$ and the operation count for $\mathscr {A}_{ 2s, M }^\mathrm {disc}$ are

  $$\begin{aligned} \mathscr {O} \left( \frac{s^2 r^{ 3/2 } \log ^{ 11/2 } M}{\log s} \right) . \end{aligned}$$

The Monte Carlo variant of $\mathscr {A}_{ 2s, M }^\mathrm {disc}$, denoted $\mathscr {A}_{ 2s, M }^\mathrm {disc, MC}$, satisfies the all of the conditions (1)–(3) simultaneously with probability $(1 - \sigma ) \in [2/3, 1)$ and has number of required samples and operation count

$$\begin{aligned} \mathscr {O} \left( s r^{ 3/2 }\log ^{ 9/2 }(M) \log \left( \frac{M}{\sigma } \right) \right) . \end{aligned}$$

Proof

All notation in this proof matches that in [27] (in particular, we use f to denote the one-dimensional function in place of a in the theorem statement and $N = 2M + 1$). We begin by substituting the $2\pi$-periodic gaussian filter given in (3) on page 756 with the 1-periodic gaussian and associated Fourier transform

$$\begin{aligned} g(x) = \frac{1}{c_1} \sum _{ n = -\infty }^\infty \mathbb { e }^{ - \frac{(2\pi )^2(x - n)^2}{2 c_1^2} },\quad {\hat{g}}_\omega = \frac{1}{\sqrt{ 2\pi }} \mathbb { e }^{ - \frac{c_1^2 \omega ^2}{2} }. \end{aligned}$$

Note then that all results regarding the Fourier transform remain unchanged, and since this 1-periodic gaussian is a just a rescaling of the $2\pi$-periodic one used in [27], the bound in [27, Lemma 1] holds with a similarly compressed gaussian, that is, for all $x \in \left[ -\frac{1}{2}, \frac{1}{2} \right]$

$$\begin{aligned} g(x) \le \left( \frac{3}{c_1} + \frac{1}{\sqrt{ 2\pi }} \right) \mathbb { e }^{ -\frac{ \left( 2\pi x \right) ^2 }{2 c_1^2} }. \end{aligned}$$

(9)

Analogous results up to and including [27, Lemma 10] for 1-periodic functions then hold straightforwardly.

Assuming that our signal measurements ${{\mathbf {f}}}= (f(y_j))_{ j = 0 }^{ 2M } = (f( \frac{j}{N} ))_{ j = 0 }^{ 2M }$ are corrupted by some discrete noise $\varvec{\mu } = (\mu _j)_{ j = 0 }^{ 2M }$, we consider for any $x \in \mathbb {T}$ a similar bound to [27, Lemma 10]. Here, $j' := \arg \min _{ j }| x - y_j|$ and $\kappa := \left\lceil \gamma \ln N \right\rceil + 1$ for some $\gamma \in \mathbb { R }^+$ to be determined. Then,

$$\begin{aligned}&\left| \frac{1}{N} \sum _{ j = 0 }^{ 2M } f(y_j) g(x - y_j) - \frac{1}{N} \sum _{ j = j' - \kappa }^{ j' + \kappa } (f(y_j) + \mu _j) g(x - y_j) \right| \\&\qquad \le \frac{1}{N} \left| \sum _{ j = 0 }^{ 2M } f(y_j) g( x-y_j ) - \sum _{ j = j' - \kappa }^{ j'+\kappa } f(y_j) g(x - y_j) \right| + \frac{1}{N} \left| \sum _{ j = j' - \kappa }^{ j' + \kappa } \mu _j g(x - y_j) \right| \\&\qquad \le \frac{1}{N} \left| \sum _{ j = 0 }^{ 2M } f(y_j) g( x-y_j ) - \sum _{ j = j' - \kappa }^{ j'+\kappa } f(y_j) g(x - y_j) \right| + \frac{\Vert \varvec{\mu }\Vert _\infty }{N} \sum _{ k = - \kappa }^{ \kappa } g(x - y_{ j' + k }) \end{aligned}$$

We bound the first term in this sum by a direct application of [27, Lemma 10]; however, we take this opportunity to reduce the constant in the bound given there. In particular, bounding this term by the final expression in the proof of [27, Lemma 10] and using our implicit assumption that $36 \le N$, we have

$$\begin{aligned} \begin{aligned}&\left| \frac{1}{N} \sum _{ j = 0 }^{ 2M } f(y_j) g(x - y_j) - \frac{1}{N} \sum _{ j = j' - \kappa }^{ j' + \kappa } (f(y_j) + \mu _j) g(x - y_j) \right| \\&\qquad \qquad \le \left( \frac{3}{\sqrt{ 2\pi }} + \frac{1}{2\pi } \sqrt{ \frac{\ln 36}{36} } \right) \Vert \mathbf {f} \Vert _\infty N^{ -r } + \frac{1}{N} \Vert \varvec{\mu } \Vert _\infty \sum _{ k = - \kappa }^{ \kappa } g(x - y_{ j' + k }). \end{aligned} \end{aligned}$$

(10)

We now work on bounding the second term. First note that for all $k \in [-\kappa , \kappa ] \cap \mathbb { Z }$,

$$\begin{aligned} g(x-y_{ j' \pm k })&= g\left( x - y_{ j' } \pm \frac{k}{N}\right) . \end{aligned}$$

Assuming without loss of generality that $0 \le x - y_{ j' }$, we can bound the nonnegatively indexed summands by (9) as

$$\begin{aligned} g\left( x - y_{ j' } + \frac{k}{N}\right)&\le \left( \frac{3}{c_1} + \frac{2}{\sqrt{ 2 \pi }}\right) \mathbb { e }^{ - \frac{(2 \pi )^2k^2}{2 c_1^2N^2} }. \end{aligned}$$

(11)

For the negatively indexed summands, the definition of $j' = \arg \min _j |x - y_j|$ implies that $x - y_{ j' } \le \frac{1}{2N}$. In particular,

$$\begin{aligned} x - y_{ j' } - \frac{k}{N} \le \frac{1 - 2k}{2N} < 0 \end{aligned}$$

implies

$$\begin{aligned} \left( x - y_{ j' } - \frac{k}{N} \right) ^2 \ge \frac{1 - 2k}{2N} \left( x- y_{ j' } - \frac{k}{N} \right) \ge \frac{2k - 1}{2N} \cdot \frac{k}{N} , \end{aligned}$$

giving

$$\begin{aligned} g \left( x - y_{ j' } - \frac{k}{N} \right) \le \left( \frac{3}{c_1} + \frac{2}{\sqrt{ 2\pi }} \right) \mathbb { e }^{ - \frac{(2\pi )^2k^2}{2c_1^2N^2} } \mathbb { e }^{ \frac{(2\pi )^2k}{4c_1^2N^2} }. \end{aligned}$$

(12)

We now bound the final exponential. We first recall from [27] the choices of parameters

$$\begin{aligned} c_1 = \frac{\beta \sqrt{ \ln N }}{N}, \quad \kappa = \left\lceil \gamma \ln N \right\rceil + 1, \quad \gamma = \frac{6r}{\sqrt{ 2 }\pi } = \frac{\beta \sqrt{ r}}{2 \sqrt{ \pi }}, \quad \beta = 6 \sqrt{ r } \end{aligned}$$

with $1 \le r \le \frac{N}{36}$. For $k \in [1, \kappa ] \cap \mathbb { Z }$ then,

$$\begin{aligned} \exp \left( \frac{(2\pi )^2k}{4c_1^2N^2} \right)&\le \exp \left( \frac{(2\pi )^2\kappa }{4c_1^2N^2} \right) \\&\le \exp \left( \frac{\pi ^2\left( \frac{6r \ln N}{\sqrt{ 2 }\pi } + 2\right) }{36 r \ln N} \right) \\&\le \exp \left( \frac{\pi }{6\sqrt{ 2 }} + \frac{\pi ^2}{18r \ln N} \right) \\&\le \exp \left( \frac{\pi }{6\sqrt{ 2 }} + \frac{\pi ^2}{18 \ln 36} \right) =: A. \end{aligned}$$

Combining this with our bounds for the nonnegatively indexed summands (11) and the negatively indexed summands (12), we have

$$\begin{aligned} \frac{1}{N} \sum _{ k = -\kappa }^\kappa g(x - y_{ j'+ k }) \le \left( \frac{3}{\beta \sqrt{ \ln N }} + \frac{1}{N \sqrt{ 2 \pi }}\right) \left( 1 + (1 + A) \sum _{ k = 1 }^\kappa \mathbb { e }^{ - \frac{(2\pi )^2 k^2}{2 \beta ^2 \ln N} } \right) \end{aligned}$$

Expressing the final sum as a truncated lower Riemann sum and applying a change of variables on the resulting integral, we have

$$\begin{aligned} \sum _{ k = 1 }^\kappa \mathbb { e }^{ - \frac{(2\pi )^2 k^2}{2 \beta ^2 \ln N} } \le \frac{\beta \sqrt{ \ln N }}{\sqrt{ 2 }\pi } \int _{ 0 }^{ \infty } e^{ -x^2 } \,d x = \frac{\beta \sqrt{ \ln N }}{2 \sqrt{ 2 \pi }}. \end{aligned}$$

Making use of our parameter values from [27], and the fact that $1 \le r \le \frac{N}{36}$,

$$\begin{aligned} \frac{1}{N} \sum _{ k = -\kappa }^\kappa g(x - y_{ j'+ k })\le & {} \left( \frac{3}{\beta \sqrt{ \ln N }} + \frac{1}{N \sqrt{ 2 \pi }}\right) \left( 1 + \frac{1 + A}{2 \sqrt{ 2 \pi }} \beta \sqrt{ \ln N } \right) \nonumber \\\le & {} \frac{3}{6 \sqrt{ \ln 36 }} + \frac{3(1 + A)}{2 \sqrt{ 2 \pi }} + \frac{1}{36 \sqrt{ 2\pi }} + \frac{1+A}{4\pi } \sqrt{ \frac{\ln 36}{36} } \nonumber \\< & {} 2. \end{aligned}$$

(13)

With our revised bound for (10) above, we reprove [27, Theorem 4] to estimate $g *f$ by the truncated discrete convolution with noisy samples. In particular, we apply [27, Theorem 3], (10), (9), and finally our same assumption that $1 \le r \le \frac{N}{36}$ to obtain

$$\begin{aligned}&\left| (g * f) (x) - \frac{1}{N} \sum _{ j = j' - \left\lceil \frac{6r}{\sqrt{ 2 } \pi } \ln N \right\rceil - 1 }^{ j' + \left\lceil \frac{6r}{\sqrt{ 2 }\pi } \ln N \right\rceil + 1 } (f(y_j) + \mu _j) g(x - y_j) \right| \\&\qquad \le \frac{N^{ 1 - r }}{6 \sqrt{ r } \sqrt{ \ln N }} \Vert \mathbf {f} \Vert _\infty N^{ -r } + \left( \frac{3}{\sqrt{ 2\pi }} + \frac{1}{2\pi } \sqrt{ \frac{\ln 36}{36} } \right) \Vert \mathbf {f} \Vert _\infty N^{ -r } + 2 \Vert \varvec{\mu } \Vert _\infty \\&\qquad \le \left( \frac{1}{6 \sqrt{ \ln 36 }} + \frac{3}{\sqrt{ 2\pi }} + \frac{1}{2\pi } \sqrt{ \frac{\ln 36}{36} } \right) \frac{ \Vert \mathbf {f} \Vert _\infty }{N^r} + 2 \Vert \varvec{\mu } \Vert _\infty < 2 \left( \frac{\Vert \mathbf {f} \Vert _\infty }{N^r} + \Vert \varvec{\mu }\Vert _\infty \right) . \end{aligned}$$

Replacing all references of $3 \Vert {{\mathbf {f}}}\Vert _{ \infty } N^{ -r }$ by $2( \Vert {{\mathbf {f}}}\Vert _\infty N^{ -r } + \Vert \varvec{\mu } \Vert _\infty )$ in the remainder of the steps up to proving [27, Theorem 5] gives the desired noise robustness (with a slightly improved constant).

Using the revised error estimates of the nonequispaced algorithm from Theorem 1 and redefining $\delta = 3 (\Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}}_s^\mathrm {opt} \Vert _1 / 2s + 2( \Vert {{\mathbf {f}}}\Vert _\infty N^{ -r } + \Vert \varvec{ \mu } \Vert _\infty ))$ as in the proof of [27, Theorem 5] (which also contains the proof of property (2)), the discretization algorithm [27, Algorithm 1] will produce candidate Fourier coefficient approximations in lines 9 and 12 corresponding to every $|{\hat{f}}_\omega | \ge (4 + 2 \sqrt{ 2 }) \delta$ in place of $4\delta$ in Theorem 1. The exact same argument as in the proof of Theorem 1 then applies to the selection of the 2s-largest entries of this approximation with the revised threshold values and error bounds to give properties (1) and (3).

In detail, [27, Lemma 13] and the discussion right after its statement gives that property (2) holds for any approximate coefficient with frequency recovered throughout the algorithm (which, for the purposes of the following discussion, we will store in $\mathbf {x}$ rather than ${\hat{R}}$ defined in [27, Algorithm 1]), not just those in the final output $\mathbf {v} := \mathbf {x}_s^{ \mathrm {opt} }$. Additionally, by the same lemma and our revised bounds from Theorem 1, any frequency $\omega \in [N]$ satisfying $|f_\omega | > (4 + 2 \sqrt{ 2 })\delta$ will have an associated coefficient estimate in $\mathbf {x}$.

By Lemma 4, $| \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta } | \le 2s = | {{\,\mathrm{supp}\,}}(\mathbf {v}) |$, and so if $\omega \in \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta } \setminus {{\,\mathrm{supp}\,}}( \mathbf {v} )$, there exists some $\omega ' \in {{\,\mathrm{supp}\,}}(\mathbf {v}) \setminus \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta }$ such that $v_{\omega '}$ took the place of $v_{ \omega }$ in $\mathscr {S}$. In particular, this means that $|x_{ \omega ' }| \ge |x_{ \omega }|$, $|{\hat{f}}_{ \omega ' }| \le (4 + 2 \sqrt{ 2 })\delta$, and $|{\hat{f}}_{ \omega }| > (4 + 2 \sqrt{ 2 }\delta )$. Thus,

$$\begin{aligned} (4 + 2 \sqrt{ 2 })\delta + \sqrt{ 2 }\delta > |{\hat{f}}_{ \omega ' }| + \sqrt{ 2 } \delta \ge |x_{ \omega ' }| \ge |x_\omega | \ge |{\hat{f}}_{ \omega }| - \sqrt{ 2 }\delta , \end{aligned}$$

implying that $|{\hat{f}}_\omega | \le 4(1 + \sqrt{ 2 })\delta$ and therefore proving (1).

Finally, to prove (3), we use Lemma 4, and consider

$$\begin{aligned} \Vert {\hat{{{\mathbf {f}}}}} - \mathbf {v} \Vert _2&\le \Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}} |_{ {{\,\mathrm{supp}\,}}(\mathbf {v}) } \Vert _2 - \Vert ( {\hat{{{\mathbf {f}}}}} - \mathbf {v} )|_{ {{\,\mathrm{supp}\,}}(v) } \Vert _2 \\&\le \Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}}|_{ \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta } \cap {{\,\mathrm{supp}\,}}(\mathbf {v}) } \Vert _2+ \sqrt{ 2 }\delta \sqrt{ 2s } \\&\le \Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}}|_{ \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta } } \Vert _2 + \Vert {\hat{{{\mathbf {f}}}}}|_{ \mathscr {S}_{ (4 + 2 \sqrt{ 2 })\delta } \setminus {{\,\mathrm{supp}\,}}(\mathbf {v}) }\Vert _2 + 2 \delta \sqrt{ s } \\&\le \Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}}_{ 2s }^\mathrm {opt}\Vert _2 + (4 + 2 \sqrt{ 2 })\delta \sqrt{ 2s } + 4(1 + \sqrt{ 2 })\delta \sqrt{ 2s } + 2 \delta \sqrt{ s } \\&\le \Vert {\hat{{{\mathbf {f}}}}} - {\hat{{{\mathbf {f}}}}}_{ 2s }^\mathrm {opt} \Vert _2 + (14 + 8 \sqrt{ 2 }) \delta \sqrt{ s } \end{aligned}$$

which finishes the proof. $\square$