Theoretical Guarantees for Phylogeny Inference from Single-Cell Lineage Tracing

CRISPR-Cas9 lineage tracing technologies have emerged as a powerful tool for investigating develop-ment in single-cell contexts, but exact reconstruction of the underlying clonal relationships in experiment is plagued by data-related complications. These complications are functions of the experimental parameters in these systems, such as the Cas9 cutting rate, the diversity of indel outcomes, and the rate of missing data. In this paper, we develop two theoretically grounded algorithms for reconstruction of the underlying phylogenetic tree, as well as asymptotic bounds for the number of recording sites necessary for exact recapitulation of the ground truth phylogeny at high probability. In doing so, we explore the relationship between the problem difficulty and the experimental parameters, with implications for experimental design. Lastly, we provide simulations validating these bounds and showing the empirical performance of these algorithms. Overall, this work provides a first theoretical analysis of phylogenetic reconstruction in the CRISPR-Cas9 lineage tracing technology.

Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

A Modified Exact Reconstruction Algorithm to Determine the Complex Permittivity Perturbations of a Cancer Affected Biological Target using Microwave Tomography Technique

Microwave Tomography Technique (MTT) is an emerging technology that is showing its effectiveness in detecting Cancer at early stage. Due to absolute random and non-deterministic characteristics of Cancer cells, more advancements are required in MTT to accurately detect the presence as well as the location of the affected region. Considering this fundamental issue, in this paper, we have proposed a modified Exact Reconstruction Algorithm (mERA) which is capable enough to provide a detailed analysis of all kinds of complex dielectric perturbations of a cancer affected biological target. In MTT, the detection of presence of a cancerous tumor inside any organ of human body has been done using different image reconstruction algorithms. On the other hand, this algorithm uses a selective data segregation mechanism to generate the perturbed complex cell permittivities of the affected organ tissues. Through this study, it has also been verified that how efficiently our proposed approach can able to detect all types of dielectric variations that may be large (20%), small (5%), positive or may be negative and even in a mixed kind of scenario where affected cells possess the mixture of all types of perturbations simultaneously. As cancerous cell shows peculiar behaviour inside human body and its nature varies from person to person and even in-between different stages (stage 1, stage 2, stage 3, stage 4) of cancer, the algorithm is designed in such a fashion that it can able to detect the presence of tumor considering all such possibilities into account. The results validate its high accuracy and effectiveness in the field of cancer diagnosis.

A Novel Sparse Data Reconstruction Algorithm for Dynamically Detect and Adjust Signal Sparsity

This paper proposed a novel algorithm which is called the joint step-size matching pursuit algorithm (JsTMP) to solve the issue of calculating the unknown signal sparsity. The proposed algorithm falls into the general category of greedy algorithms. In the process of iteration, this method can adjust the step size and correct the indices of the estimated support that were erroneously selected in a dynamical way. And it uses the dynamical step sizes to increase the estimated sparsity level when the energy of the residual is less than half of that of the measurement vectory. The main innovations include two aspects: 1) The high probability of exact reconstruction, comparable to other classical greedy algorithms reconstruct arbitrary spare signal. 2) The sinh() function is used to adjust the right step with the value of the objective function in the late iteration. Finally, by following this approach, the simulation results show that the proposed algorithm outperforms state of- the-art similar algorithms used for solving the same problem.

Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $$f(t) = \sum _{j=1}^{K} \gamma _{j} \, \cos (2\pi a_{j} t + b_{j})$$ f ( t ) = ∑ j = 1 K γ j cos ( 2 π a j t + b j ) , where the frequency parameters $$a_{j} \in {\mathbb {R}}$$ a j ∈ R (or $$a_{j} \in {\mathrm i} {\mathbb {R}}$$ a j ∈ i R ) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients of f with regard to a fixed interval (0, P) with $$P>0$$ P > 0 . Even though all terms of f may be non-P-periodic, our reconstruction method requires at most $$2K+2$$ 2 K + 2 Fourier coefficients $$c_{n}(f)$$ c n ( f ) to recover all parameters of f. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $$K+1$$ K + 1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and $$L \ge 2K+2$$ L ≥ 2 K + 2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

A Literature Review of Rapid Prototyping and Patient Specific Implants for the Treatment of Orbital Fractures

Post-traumatic reconstruction of the orbit can pose a challenge due to inherent intraoperative problems. Intra-orbital adipose tissue is difficult to manipulate and retract making visualization of the posterior orbital contents difficult. Rapid prototyping (RP) is a cost-effective method of anatomical model production allowing the surgeon to produce a patient specific implant (PSI) which can be pre-surgically adapted to the orbital defect with exact reconstruction. Intraoperative imaging allows immediate assessment of reconstruction at the time of surgery. Utilization and combination of both technologies improves accuracy of reconstruction with orbital implants and reduces cost, surgical time, and the rate of revision surgery.

Non-stationary dynamic system characteristics recovery from three test signals

Algorithms of exact restoration in an analytical form of dynamic characteristics of non-stationary dynamic systems are constructed. Non-stationary continuous dynamical systems modeled by Volterra integral equations of the first kind and non-stationary discrete dynamical systems modeled by discrete analogues of Volterra integral equations of the first kind are considered.The article consists of an introduction and three sections: 1) The exact restoration of the dynamic characteristics of continuous systems, 2) The restoration of the transition characteristics of discrete systems, 3) Conclusions. The introduction provides a statement of the problem and provides an overview of dynamical systems for which algorithms for exact reconstruction in ananalytical form of the impulse response (in the case of continuous systems) and the transition characteristic (in the case of discrete systems) are constructed. In the first section, the algorithm is constructed for the exact reconstruction of the impulse response of an non-stationary continuous dynamic system from three interconnected input signals. The first signal may be arbitrary, the second and third signals are associated with the first signal by integral operator. The exact formula for the Laplace transform of the impulse response, represented by an algebraic expression from the Laplace transform of the system output signals, is given. A model example illustrating the effectiveness of the algorithm is given. The practical application of the presented algorithm isdiscussed. In the second section, an algorithm is constructed for the exact reconstruction of the transition response of a non-stationary discrete dynamical system from three input signals that are interconnected. The first signal may be arbitrary, the second and third signals are associated with the first summing operator. The exact formula of the Z-transform of the transition characteristic is presented, which is represented by an algebraic expression from the Z-transform of the system output signals. A model example is given. The “Conclusions” section provides a summary of the results presented in the article and describes the dynamic systems to which the proposed algorithms can be extended.