On the worst-case error of least squares algorithms for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e27" altimg="si11.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>-approximation with high probability

Ullrich, Mario

doi:10.1016/j.jco.2020.101484

“…Let us conclude with a few topics for future research. While this paper was under review, Theorem 1 has already been extended to the case of complex-valued functions and non-injective operators id : H → L 2 in [7], including explicit values for the constants c and C, see also [21]. It remains open to generalize our results to non-Hilbert space settings.…”

Section: Mathematics Subject Classification 41a25 • 41a46 • 60b20 • Smentioning

confidence: 94%

Function Values Are Enough for $$L_2$$-Approximation

Krieg

¹

,

Ullrich

²

2020

Self Cite

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We study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

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“…which represents a slightly weaker bound. Further progress (explicit constants, consequences for numerical integration) has been given in Kämmerer et al [13] and in M.Ullrich [41] as well as Moeller and T.Ullrich [24] for the control of the failure probability.…”

Section: Remark 62 (I)mentioning

confidence: 99%

“…In addition, the result in this paper partly relies on this random sampling strategy according to a distribution built upon spectral properties of the embedding. The advantage of the pure random strategy in connection with a log(n)-oversampling is the fact that the failure probability decays polynomially in n which has been recently shown by M.Ullrich [41] and, independently, by Moeller together with the third named author [24]. In other words, although this approach incorporates a probabilistic ingredient, the failure probability is controlled and the algorithm may be implemented.…”

Section: Introductionmentioning

confidence: 99%

A New Upper Bound for Sampling Numbers

Nagel

¹

,

Schäfer

²

,

Ullrich

³

2021

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We provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) n ∈ N associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which are specified in the paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ g n 2 ≤ C log ( n ) n ∑ k ≥ ⌊ c n ⌋ σ k 2 , n ≥ 2 , where $$(\sigma _k)_{k\in \mathbb {N}}$$ ( σ k ) k ∈ N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H ( K ) → L 2 ( D , ϱ D ) . The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\text {mix}}(\mathbb {T}^d)$$ H mix s ( T d ) in $$L_2(\mathbb {T}^d)$$ L 2 ( T d ) with $$s>1/2$$ s > 1 / 2 . We obtain the asymptotic bound $$\begin{aligned} g_n \le C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, \end{aligned}$$ g n ≤ C s , d n - s log ( n ) ( d - 1 ) s + 1 / 2 , which improves on very recent results by shortening the gap between upper and lower bound to $$\sqrt{\log (n)}$$ log ( n ) . The result implies that for dimensions $$d>2$$ d > 2 any sparse grid sampling recovery method does not perform asymptotically optimal.

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“…The following result already improves on the result given in [11,13] in several directions. The theorem works under less restrictive assumptions, the constants are improved and, last but not least, the failure probability decays polynomially in n. We would like to point that, while preparing this manuscript, Ullrich [29] proved a version of the next theorem with stronger requirements and different constants based on Oliveira's concentration result (see Remark 3.9). The following theorem is a reformulation of Theorem 5.2 in Sect.…”

Section: Introductionmentioning

confidence: 95%

“…In addition, we are interested in the sampling discretization of the squared L 2 -norm of such functions using n random nodes. Both problems recently gained substantial interest, see [11,13,14,[25][26][27]29], and are strongly related as we know from Wasilkowski Communicated by Deguang Han.…”

Section: Introductionmentioning

confidence: 99%

$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

Moeller

¹

,

Ullrich

²

2021

Sampl. Theory Signal Process. Data Anal.

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In this paper we study $$L_2$$ L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $$D \subset \mathbb {R}^d$$ D ⊂ R d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into $$L_2$$ L 2 ) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in n, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.

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On the worst-case error of least squares algorithms for L2-approximation with high probability

Cited by 25 publications

References 7 publications

Function Values Are Enough for $$L_2$$-Approximation

Function Values Are Enough for $$L_2$$-Approximation

A New Upper Bound for Sampling Numbers

$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

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