Abstract. We discover a new instability mechanism for short-time Fourier transform phase retrieval which yields that for any reasonable window function $\varphi$ in any dimension $d$, the local stability constant $c(f)$ defined via
$$\inf_{|\lambda|=1}\|f- \lambda g\|_{M^p(\mathbb{R}^{d})}\leq c(f)\| |V_\varphi f|-|V_\varphi g|\|_\mathcal{D}, \hspace{5mm} \forall g\in M^p(\mathbb{R}^d),$$
is infinite on a dense set of vectors for all weighted fractional Sobolev norms $\mathcal{D}$, up to the sharp maximal regularity level ensuring that the problem is well-defined. This, in particular, answers an open problem of Rathmair, who asked whether exponential concentration of the Gabor transform on $\mathbb{R}^2$ guaranteed a finite local stability constant. For the specific case of Gabor phase retrieval, we further show that there is a complementary dense set where the local stability constant on $\mathbb{R}^{2d}$ is finite.
Our results extend and complement a series of fundamental stability theorems for Gabor phase retrieval which have been proven over the last ten years. Of particular note is the work of Grohs and Rathmair, who showed that for sufficiently strong weighted Sobolev norms $\mathcal{D}$ on $\mathbb{R}^{2d}$, the local stability constant for Gabor phase retrieval is bounded by the inverse of the Cheeger constant of the flat metric conformally multiplied by $|V_\varphi f|$. As a consequence of our analysis, we determine two dense families of functions, one of which has associated Cheeger constant zero and the other strictly positive. We also revisit the stability problem for STFT phase retrieval on bounded subsets of the time-frequency plane, for more general windows, and for restricted signal classes, extending and simplifying many influential results in the literature.

Abstract. The signature kernel [9] is a recent state-of-the-art tool for analyzing high-dimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via dynamically truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE [16], our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent local Goursat PDEs by recursively propagating boundary conditions along a directed graph via topological ordering, with dynamic truncation adaptively terminating each local power series expansion when coefficients fall below machine precision, striking an effective balance between computational cost and accuracy. This method achieves substantial performance improvements over state-of-the-art approaches for computing the signature kernel, providing (a) adjustable and superior accuracy, even for time series with very high roughness; (b) drastically reduced memory requirements; and (c) scalability to efficiently handle very long time series (e.g., with up to half a million points or more) on a single GPU. These advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications that involve very long and highly volatile data.

Abstract. Phase retrieval from phaseless short-time Fourier transform (STFT) measurements is known to be inherently unstable when measurements are taken with respect to a single window. While an explicit inversion formula exists, it is useless in practice due to its instability. In this paper, we overcome this lack of stability by presenting two multi-window approaches that rely on a “good coverage” of the time-frequency plane by the ambiguity functions of the windows. The first is to use the fractional Fourier transform of a dilated Gauss function with various angles as window functions. The essential support of a superposition of the ambiguity function from such window functions is of a “daffodil shape”, which converges to a large disc as more angles are used, yielding a much broader coverage in the time-frequency domain. The second approach uses Hermite functions of various degrees as the window functions. The larger the degree, the wider the ambiguity function but with zeros on circles in the time-frequency domain. Combining Hermite functions of different degrees, we can achieve a wide coverage with zeros compensated by the essential support of the ambiguity function from other Hermite windows. Taking advantage of these multi-window procedures, we can stably perform STFT phase retrieval using the direct inversion formula.

Abstract. Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of $L^2$ norms. The focus of this paper is on stability with respect to the $L^\infty$ norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in $L^\infty$, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in $L^\infty$. Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in $L^\infty$. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.

Abstract. Nonparametric regression of stochastic processes estimates statistical relationships between multidimensional, time-dependent data without relying on specific parametric assumptions. We propose a novel approach to this classical estimation problem by using the signature transform from rough path theory to encode the information of a stochastic process into a sequence of iterated integrals, capturing its statistical properties in a time-global and hierarchical manner. Viewing statistical regression as an operator learning problem, this signature-based discretisation allows us to characterise the conditional statistical dependence of a stochastic process on another stochastic process as the solution to a convex semi-infinite linear least squares problem.
This result is based on a functional monotone class argument involving the bounded signature of the conditioning process and allows for the efficient and provably consistent nonparametric estimation of regression functions and conditional distributions for very general classes of jointly distributed stochastic processes as solutions to convex optimisation problems. The structural insights of this approach are summarised in two universally consistent regression estimators that are computable with practical algorithms and supported by broad theoretical guarantees.

Abstract. Recently, there has been significant interest in operator learning, i.e. learning mappings between infinite-dimensional function spaces. This has been particularly relevant in the context of learning partial differential equations from data. However, it has been observed that proposed models may not behave as operators when implemented on a computer, questioning the very essence of what operator learning should be. We contend that in addition to defining the operator at the continuous level, some form of continuous-discrete equivalence is necessary for an architecture to genuinely learn the underlying operator, rather than just discretizations of it. To this end, we propose to employ frames, a concept in applied harmonic analysis and signal processing that gives rise to exact and stable discrete representations of continuous signals. Extending these concepts to operators, we introduce a unifying mathematical framework of Representation equivalent Neural Operator (ReNO) to ensure operations at the continuous and discrete level are equivalent. Lack of this equivalence is quantified in terms of aliasing errors. We analyze various existing operator learning architectures to determine whether they fall within this framework, and highlight implications when they fail to do so.

Abstract. Although very successfully used in conventional machine learning, convolution based neural network architectures — believed to be inconsistent in function space — have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning.

Abstract. X-ray grating interferometry CT (GI-CT) is an emerging imaging modality which provides three complementary contrasts that could increase the diagnostic content of clinical breast CT: absorption, phase, and dark-field. Yet, reconstructing the three image channels under clinically compatible conditions is challenging because of severe ill-conditioning of the tomographic reconstruction problem. In this work we propose to solve this problem with a novel reconstruction algorithm that assumes a fixed relation between the absorption and the phase-contrast channel to reconstruct a single image by automatically fusing the absorption and phase channels. The results on both simulations and real data show that, enabled by the proposed algorithm, GI-CT outperforms conventional CT at a clinical dose.

Abstract. We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients
$$
\{\lvert \mathcal{W}_{\phi_i} f(\alpha^{m}\beta n,\alpha^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\}
$$
for every $\alpha>1,\beta>0$ with $\beta\ln(\alpha)\leq 4\pi/(1+4p)$, $p>0$, when the three wavelets $\phi_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform. For complex-valued signals we find that this is not possible for any choice of the parameters $\alpha>1,\beta>0$ and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.

Abstract. As interest in deep neural networks (DNNs) for image reconstruction tasks grows, their reliability has been called into question (Antun et al., 2020; Gottschling et al., 2020). However, recent work has shown that compared to total variation (TV) minimization, they show similar robustness to adversarial noise in terms of $\ell^2$-reconstruction error (Genzel et al., 2022). We consider a different notion of robustness, using the $\ell^\infty$-norm, and argue that localized reconstruction artifacts are a more relevant defect than the $\ell^2$-error. We create adversarial perturbations to undersampled MRI measurements which induce severe localized artifacts in the TV-regularized reconstruction. The same attack method is not as effective against DNN based reconstruction. Finally, we show that this phenomenon is inherent to reconstruction methods for which exact recovery can be guaranteed, as with compressed sensing reconstructions with $\ell^1$- or TV-minimization.

Abstract. For every lattice $\Lambda$, we construct functions which are arbitrarily close to the Gaussian, do not agree up to global phase but have Gabor transform magnitudes agreeing on $\Lambda$. Additionally, we prove that the Gaussian can be uniquely recovered (up to global phase) in $L^2(\mathbb{R})$ from Gabor magnitude measurements on a sufficiently fine lattice. These two facts give evidence for the existence of functions which break uniqueness from samples without affecting stability. We prove that a uniform bound on the local Lipschitz constant of the signals is not sufficient to restore uniqueness in sampled Gabor phase retrieval and more restrictive a priori knowledge of the functions is necessary. With this, we show that there is no direct connection between uniqueness from samples and stability in Gabor phase retrieval. Finally, we provide an intuitive argument about the connection between directions of instability in phase retrieval and Laplacian eigenfunctions associated to small eigenvalues.

Abstract. We study the problem of phase retrieval in which one aims to recover a function $f$ from the magnitude of its wavelet transform $|W_\psi f|$. We consider bandlimited functions and derive new uniqueness results for phase retrieval, where the wavelet itself can be complex-valued. In particular, we prove the first uniqueness result for the case that the wavelet $\psi$ has a finite number of vanishing moments. In addition, we establish the first result on unique reconstruction from samples of the wavelet transform magnitude when the wavelet coefficients are complex-valued.

Abstract. Breast cancer remains the most prevalent malignancy in women in many countries around the world, thus calling for better imaging technologies to improve screening and diagnosis. Grating interferometry (GI)-based phase contrast X-ray CT is a promising technique which could make the transition to clinical practice and improve breast cancer diagnosis by combining the high three-dimensional resolution of conventional CT with higher soft-tissue contrast. Unfortunately though, obtaining high-quality images is challenging. Grating fabrication defects and photon starvation lead to high noise amplitudes in the measured data. Moreover, the highly ill-conditioned differential nature of the GI-CT forward operator renders the inversion from corrupted data even more cumbersome. In this paper, we propose a novel regularized iterative reconstruction algorithm with an improved tomographic operator and a powerful data-driven regularizer to tackle this challenging inverse problem. Our algorithm combines the L-BFGS optimization scheme with a data-driven prior parameterized by a deep neural network. Importantly, we propose a novel regularization strategy to ensure that the trained network is non-expansive, which is critical for the convergence and stability analysis we provide. We empirically show that the proposed method achieves high quality images, both on simulated data as well as on real measurements.

Abstract. Inverse (ill-posed) problems appear in many applications such as medical imaging, astronomy, seismic imaging, nondestructive testing, signal processing, etc. Typically, these problems cannot be solved by conventional methods as they suffer from instabilities and regularization is required. This chapter has evolved from a mini-course taught at the Summer School on Applied Harmonic Analysis and Machine Learning at the University of Genoa in 2019. It offers an overview of the theory of inverse problems and discusses three ill-posed problems that have been studied rather recently in the literature: limited data reconstruction in computerized tomography, phase retrieval, and image classification with DNNs. The selection highlights that for modern problems, the usefulness of standard theory of regularization can be limited.

Abstract. We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(\mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions have good concentration in both time and frequency and can be constructed to be real-valued for rectangular lattices.

Abstract. We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this short note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples in a real-valued setting.

Abstract. Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that, in the deterministic setting, phase retrieval from frame coefficients is always unstable in infinite dimensional Hilbert spaces [5] and possibly severely ill-conditioned in finite dimensional Hilbert spaces [5].
Recently, it was also shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable when one is willing to accept a more relaxed semi-global stability regime [1].
We present first evidence that this semi-global stability regime allows one to do phase retrieval from measurements induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales linearly in the space dimension. To this end, we utilise well-known reconstruction formulae which have been used repeatedly in recent years [4], [6-8].

Abstract. The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, the problem is always stable in finite-dimensional settings. A prominent example of such a phase retrieval problem is the recovery of a signal from the modulus of its Gabor transform. In this paper, we study Gabor phase retrieval and ask how the stability degrades on a natural family of finite-dimensional subspaces of the signal domain $L^2(\mathbb{R})$. We prove that the stability constant scales at least quadratically exponentially in the dimension of the subspaces. Our construction also shows that typical priors such as sparsity or smoothness promoting penalties do not constitute regularization terms for phase retrieval.

Abstract. Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered matrix product $A^m B^n$? For example, is
$$ \| AABAABABB \| \leq \| AAAAABBBB \| ?$$
Drury [10] has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices $A,B$. However, the $1$-parameter family of counterexamples Drury constructs for these characterizations is comprised of $3 \times 3$ matrices, and thus as stated the characterization applies only for $N \times N$ matrices with $N \geq 3$.
In contrast, we prove that for $2 \times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N \times N$ matrices, the general rearrangement inequality holds for all disordered words, for most $A,B$ (in a sense of full measure) that are sufficiently small perturbations of the identity.

Abstract. While deep neural networks have proven to be a powerful tool for many recognition and classification tasks, their stability properties are still not well understood. In the past, image classifiers have been shown to be vulnerable to so-called adversarial attacks, which are created by additively perturbing the correctly classified image.
In this paper, we propose the ADef algorithm to construct a different kind of adversarial attack created by iteratively applying small deformations to the image, found through a gradient descent step. We demonstrate our results on MNIST with a convolutional neural network and on ImageNet with Inception-v3 and ResNet-101.

Abstract. The problem of phase retrieval is to determine a signal $f\in \mathcal{H}$, with $
\mathcal{H}$ a Hilbert space, from intensity measurements
$|F(\omega)|:=|\langle f , \varphi_\omega\rangle|$ associated with a measurement system $(\varphi_\omega)_{\omega\in \Omega}\subset \mathcal{H}$. Such problems can be seen in a wide variety of applications, ranging from X-ray crystallography, microscopy to audio processing and deep learning algorithms and accordingly, a large body of literature treating the mathematical and algorithmic solution of phase retrieval problems has emerged in recent years.

Recent work [9,3] has shown that, whenever $\mathcal{H}$ is infinite-dimensional, phase retrieval is never uniformly stable, and that, although it is always stable in the finite dimensional setting, the stability deteriorates severely in the dimension of the problem [9]. Any finite dimensional approximation of an infinite dimensional problem has to take into account this phenomenon which makes one wonder whether phase retrieval is even advisable in these situations.

On the other hand, all observed instabilities are of a certain type: they occur whenever the function $|F|$ of intensity measurements is concentrated on disjoint sets $D_j\subset \Omega$, i.e., when $F= \sum_{j=1}^k F_j$ where $F_j$ is concentrated on $D_j$ (and $k \geq 2$). Indeed, it is easy to see that intensity measurements of any function $\sum_{j=1}^k e^{\i \alpha_j} F_j$ will be close to those of $F$ while the functions themselves need not be close at all.

Motivated by these considerations we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing $F$ up to a phase factor that is not global, but that can be different for each of the subsets $D_j$, i.e., recovering $F$ up to the equivalence
$F \sim \sum_{j=1}^k e^{\i \alpha_j} F_j.$

We present concrete applications (for example in X-ray diffraction imaging or audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance if the measurement system is a Gabor frame or a frame of Cauchy wavelets.

Abstract. We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [9], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame.

We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{R}$) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when $\mathbb{K}=\mathbb{R}$ the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [5], which was originally formulated for finite-dimensional spaces: for the case $\mathbb{K}=\mathbb{R}$ the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

Abstract. In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions $\{\psi_{\lambda}\}_{\lambda\in \Lambda}\subset L^2(\mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-valued function $f \in L^2(\mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${\{|f \ast \psi_\lambda|\}_{\lambda \in \Lambda}}$.
We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $\infty$, it suffices to know $|f \ast \psi_\lambda|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor).
We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $L^2(\mathbb{R}^d)$, $d=1,2$, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

Abstract. In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in L^2(\mathcal F)$, where $\mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $\mathcal G$ that only overlaps but does not cover $\mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $\mathcal F \cap \mathcal G$. We show that with this restriction and by assuming prior knowledge on the $L^2$ norm or on the variation of $f$, better stability with Hölder continuity (typical for mildly ill-posed problems) can be obtained.

Abstract. Given two intervals $I, J \subset \mathbb{R}$ that are either disjoint or overlap, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. This problem has a unique answer (the nullspace is trivial) but is severely ill-posed. We isolate the difficulty and show that by restricting $f$ to functions having their variation bounded, reconstruction becomes stable. In particular, for functions $f \in H^1(I)$, we show that
$$ \|Hf\|_{L^2(J)} \geq c_1 \exp{\left(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\right)} \| f \|_{L^2(I)} ,$$
for some constants $c_1, c_2 > 0$ depending only on $I, J$. This inequality is sharp, however, it remains an open problem whether $\|f_x\|_{L^2(I)}$ can be replaced by $\|f_x\|_{L^1(I)}$.

Abstract. The truncated Hilbert transform with overlap $H_T$ is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). Recent work [1] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which
the singular values approach zero and one. In this paper, we exploit the property that $H_T$ commutes with a second-order differential operator $L_S$ and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of $H_T$.

Abstract. We study a restriction of the Hilbert transform as an operator $H_T$ from $L^2(a_2,a_4)$ to $L^2(a_1,a_3)$ for real numbers $a_1 < a_2 < a_3 < a_4$. The operator $H_T$ arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions $f$ supported on compact intervals $[a_2,a_4]$ from its Hilbert transform measured on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. We show that the inversion of $H_T$ is ill-posed, which is why we investigate the spectral properties of $H_T$. We relate the operator $H_T$ to a self-adjoint two-interval Sturm-Liouville problem, for which we prove that the spectrum is discrete. The Sturm-Liouville operator is found to commute with $H_T$, which then implies that the spectrum of $H_T^* H_T$ is discrete. Furthermore, we express the singular value decomposition of $H_T$ in terms of the solutions to the Sturm-Liouville problem. The singular values of $H_T$ accumulate at both 0 and 1, implying that $H_T$ is not a compact operator. We conclude by illustrating the properties obtained for $H_T$ numerically.

Abstract. The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by $N$ landmark points, chosen to be homologous, as far as possible, and the Procrustes distance is then computed as $\inf_{R}\,\sum_{j=1}^N\, \|Rx_j-x'_j\|^2$, where the minimization is over all Euclidean transformations, and the correspondences $x_j \leftrightarrow x'_j$ are picked in an optimal way. The discrete Procrustes distance has the drawback that each shape is represented by only a finite number of points, which may not capture all the geometric aspects of interest; a need has been expressed for alternatives that are still easy to compute. We propose in this paper the concept of continuous Procrustes distance, and prove that it provides a true metric for two-dimensional surfaces embedded in three dimensions. We also propose an efficient algorithm to calculate approximations to this new distance.