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Genome sequencing of cancers often reveals mosaics of different subclones present in the same tumour1–3. Although these are believed to arise according to the principles of somatic evolution, the exact spatial growth patterns and underlying mechanisms remain elusive4,5. Here, to address this need, we developed a workflow that generates detailed quantitative maps of genetic subclone composition across whole-tumour sections. These provide the basis for studying clonal growth patterns, and the histological characteristics, microanatomy and microenvironmental composition of each clone. The approach rests on whole-genome sequencing, followed by highly multiplexed base-specific in situ sequencing, single-cell resolved transcriptomics and dedicated algorithms to link these layers. Applying the base-specific in situ sequencing workflow to eight tissue sections from two multifocal primary breast cancers revealed intricate subclonal growth patterns that were validated by microdissection. In a case of ductal carcinoma in situ, polyclonal neoplastic expansions occurred at the macroscopic scale but segregated within microanatomical structures. Across the stages of ductal carcinoma in situ, invasive cancer and lymph node metastasis, subclone territories are shown to exhibit distinct transcriptional and histological features and cellular microenvironments. These results provide examples of the benefits afforded by spatial genomics for deciphering the mechanisms underlying cancer evolution and microenvironmental ecology.

Realising the full potential of novel image-based spatial transcriptomic (IST) technologies requires robust and accurate algorithms for decoding the hundreds of thousand fluorescent signals each derived from single molecules of mRNA. In this paper, we introduce PoSTcode, a probabilistic method for transcript decoding from cyclic multi-channel images, whose effectiveness is demonstrated on multiple large-scale datasets generated using different versions of the in situ sequencing protocols. PoSTcode is based on a re-parametrised matrix-variate Gaussian mixture model designed to account for correlated noise across fluorescence channels and imaging cycles. PoSTcode is shown to recover up to 50% more confidently decoded molecules while simultaneously decreasing transcript mislabeling when compared to existing decoding techniques. In addition, we demonstrate its increased stability to various types of noise and tuning parameters, which makes this new approach reliable and easy to use in practice. Lastly, we show that PoSTcode produces fewer doublet signals compared to a pixel-based decoding algorithm.

We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected axis-aligned random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. We provide theoretical guarantees under which our principal subspace estimator can attain the minimax optimal rate of convergence in polynomial time. In addition, our theory provides a more refined understanding of the statistical and computational trade-off in the problem of sparse principal component estimation, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite-sample performance., Comment: 32 pages

Subclonality is a universal feature of cancers yet how clones grow, are spatially organised, differ phenotypically or influence clinical outcome is unclear. To address this, we developed base specific in situ sequencing (BaSISS). In fixed tissues, transcripts harbouring clone-defining mutations are detected, converted into quantitative clone maps and characterised through multi-layered data integration. Applied to 8 samples from key stages of breast cancer progression BaSISS localised 1.42 million genotype informative transcripts across 4.9cm2 of tissue. Microscopic clonal topographies are shaped by resident tissue architectures. Distinct transcriptional, histological and immunological features distinguish coexistent genetic clones. Spatial lineage tracing temporally orders clone features associated with the emergence of aggressive clinical traits. These results highlight the pivotal role of spatial genomics in deciphering the mechanisms underlying cancer progression.

In this paper, we introduce a computational framework for recovering a high-resolution approximation of an unknown function from its low-resolution indirect measurements as well as high-resolution training observations by merging the frameworks of generalized sampling and functional principal component analysis. In particular, we increase the signal resolution via a data driven approach, which models the function of interest as a realization of a random field and leverages a training set of observations generated via the same underlying random process. We study the performance of the resulting estimation procedure and show that high-resolution recovery is indeed possible provided appropriate low-rank and angle conditions hold and provided the training set is sufficiently large relative to the desired resolution. Moreover, we show that the size of the training set can be reduced by leveraging sparse representations of the functional principal components. Furthermore, the effectiveness of the proposed reconstruction procedure is illustrated by various numerical examples., Comment: Accepted for publication in Springer's Advances in Computational Mathematics

We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first k derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as k increases, it grows linearly in \(k+1\) with the constant of proportionality \(1/\mathrm {e}\). Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at \(k+1\) times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval \(s+1\) bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in \(s+1\) for large s, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large s, we obtain the same result as in the case of univariate sampling with s derivatives.

In a series of recent papers (Adcock, Hansen and Poon, 2013, Appl. Comput. Harm. Anal. 45(5):3132-3167), (Adcock, Gataric and Hansen, 2014, SIAM J. Imaging Sci. 7(3):1690-1723) and (Adcock, Hansen, Kutyniok and Ma, 2015, SIAM J. Math. Anal. 47(2):1196-1233), it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of $\mathcal{O}(M(N)\log N)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from $M(N)$ Fourier samples. Therefore, due to the linear correspondences between the number of samples $M$ and number of coefficients $N$ shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data.

We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example. Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries. The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum. As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data., 24 pages

We present recently devised approach for recovery of compactly supported multivariate functions from nonuniform samples of their Fourier transforms. This is the so-called nonuniform generalized sampling (NUGS), based on a generalized sampling framework which permits an arbitrary choice of the reconstruction space and where nonuniform sampling is modeled via weighted Fourier frames. We establish a sharp sampling density which is sufficient to guarantee stable recovery, without imposing any separation condition on the sampling points. In particular for the stable NUGS recovery, we also provide sufficient sampling bandwidths in the case of one-dimensional wavelet reconstructions and show sufficient linear scaling of the sampling bandwidth and the number of wavelets.

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.

We consider the problem of recovering a compactly-supported function from a finite collection of pointwise samples of its Fourier transform taking nonuniformly. First, we show that under suitable conditions on the sampling frequencies - specifically, their density and bandwidth - it is possible to recover any such function $f$ in a stable and accurate manner in any given finite-dimensional subspace; in particular, one which is well suited for approximating $f$. In practice, this is carried out using so-called nonuniform generalized sampling (NUGS). Second, we consider approximation spaces in one dimension consisting of compactly supported wavelets. We prove that a linear scaling of the dimension of the space with the sampling bandwidth is both necessary and sufficient for stable and accurate recovery. Thus wavelets are up to constant factors optimal spaces for reconstruction.