Underdetermined Problem Research Articles

SwCAM: estimation of subtype-specific expressions in individual samples with unsupervised sample-wise deconvolution.

Complex biological tissues are often a heterogeneous mixture of several molecularly distinct cell subtypes. Both subtype compositions and subtype-specific (STS) expressions can vary across biological conditions. Computational deconvolution aims to dissect patterns of bulk tissue data into subtype compositions and STS expressions. Existing deconvolution methods can only estimate averaged STS expressions in a population, while many downstream analyses such as inferring co-expression networks in particular subtypes require subtype expression estimates in individual samples. However, individual-level deconvolution is a mathematically underdetermined problem because there are more variables than observations. We report a sample-wise Convex Analysis of Mixtures (swCAM) method that can estimate subtype proportions and STS expressions in individual samples from bulk tissue transcriptomes. We extend our previous CAM framework to include a new term accounting for between-sample variations and formulate swCAM as a nuclear-norm and ℓ2,1-norm regularized matrix factorization problem. We determine hyperparameter values using cross-validation with random entry exclusion and obtain a swCAM solution using an efficient alternating direction method of multipliers. Experimental results on realistic simulation data show that swCAM can accurately estimate STS expressions in individual samples and successfully extract co-expression networks in particular subtypes that are otherwise unobtainable using bulk data. In two real-world applications, swCAM analysis of bulk RNASeq data from brain tissue of cases and controls with bipolar disorder or Alzheimer's disease identified significant changes in cell proportion, expression pattern and co-expression module in patient neurons. Comparative evaluation of swCAM versus peer methods is also provided. The R Scripts of swCAM are freely available at https://github.com/Lululuella/swCAM. A user's guide and a vignette are provided. Supplementary data are available at Bioinformatics online.

Blind inversion of multidimensional seismic data using sequential Tikhonov and total variation regularizations

The acoustic impedance (AI) model is key data for seismic interpretation, usually obtained from its nonlinear relation with seismic reflectivity. Common approaches use initial geologic and seismic information to constrain the AI model estimation. When no accurate prior information is available, these approaches may dictate false results at some parts of the model. The regularization of ill-posed underdetermined problems requires some constraints to restrict the possible results. Available seismic inversion methods mostly use Tikhonov or total variation (TV) regularizations with some adjustments. Tikhonov regularization assumes smooth variation in the AI model, and it is incurious about the rapid changes in the model. TV allows rapid changes, and it is more stable in the presence of noisy data. In a detailed realistic earth model that AI changes gradually, TV creates a staircasing effect, which could lead to misinterpretation. This could be avoided by using TV and Tikhonov regularization sequentially in the alternating direction method of multipliers and creating the AI model. The result of implementing our algorithm, sequential Tikhonov and total variation regularizations (STTVR), on 2D synthetic and real seismic sections indicates that the smaller details in the lithologic variations are accounted for as well as the general trend. STTVR can calculate major AI variations without any additional low-frequency constraints. The temporal and spatial transition of the calculated AI in real seismic data is gradual and close to a real geologic setting.

3D denoised completion network for deep single-pixel reconstruction of hyperspectral images.

Single-pixel imaging acquires an image by measuring its coefficients in a transform domain, thanks to a spatial light modulator. However, as measurements are sequential, only a few coefficients can be measured in the real-time applications. Therefore, single-pixel reconstruction is usually an underdetermined inverse problem that requires regularization to obtain an appropriate solution. Combined with a spectral detector, the concept of single-pixel imaging allows for hyperspectral imaging. While each channel can be reconstructed independently, we propose to exploit the spectral redundancy between channels to regularize the reconstruction problem. In particular, we introduce a denoised completion network that includes 3D convolution filters. Contrary to black-box approaches, our network combines the classical Tikhonov theory with the deep learning methodology, leading to an explainable network. Considering both simulated and experimental data, we demonstrate that the proposed approach yields hyperspectral images with higher quantitative metrics than the approaches developed for grayscale images.

Multichannel adaptive deconvolution based on streaming prediction-error filter

Abstract Deconvolution mainly improves the resolution of seismic data by compressing seismic wavelets, which is of great significance in high-resolution processing of seismic data. Prediction-error filtering/least-square inverse filtering is widely used in seismic deconvolution and usually assumes that seismic data is stationary. Affected by factors such as earth filtering, actual seismic wavelets are time- and space-varying. Adaptive prediction-error filters are designed to effectively characterise the nonstationarity of seismic data by using iterative methods, however, it leads to problems such as slow calculation speed and high memory cost when dealing with large-scale data. We have proposed an adaptive deconvolution method based on a streaming prediction-error filter. Instead of using slow iterations, mathematical underdetermined problems with the new local smoothness constraints are analytically solved to predict time-varying seismic wavelets. To avoid the discontinuity of deconvolution results along the space axis, both time and space constraints are used to implement multichannel adaptive deconvolution. Meanwhile, we define the parameter of the time-varying prediction step that keeps the relative amplitude relationship among different reflections. The new deconvolution improves the resolution along the time direction while reducing the computational costs by a streaming computation, which is suitable for handling nonstationary large-scale data. Synthetic model and field data tests show that the proposed method can effectively improve the resolution of nonstationary seismic data, while maintaining the lateral continuity of seismic events. Furthermore, the relative amplitude relationship of different reflections is reasonably preserved.

Diagnosing underdetermination in stable isotope mixing models.

Stable isotope mixing models (SIMMs) provide a powerful methodology for quantifying relative contributions of several sources to a mixture. They are widely used in the fields of ecology, geology, and archaeology. Although SIMMs have been rapidly evolved in the Bayesian framework, the underdetermination of mixing space remains problematic, i.e., the estimated relative contributions are incompletely identifiable. Here we propose a statistical method to quantitatively diagnose underdetermination in Bayesian SIMMs, and demonstrate the applications of our method (named β-dependent SIMM) using two motivated examples. Using a simulation example, we showed that the proposed method can rigorously quantify the expected underdetermination (i.e., intervals of β-dependent posterior) of relative contributions. Moreover, the application to the published field data highlighted two problematic aspects of the underdetermination: 1) ordinary SIMMs was difficult to quantify underdetermination of each source, and 2) the marginal posterior median was not necessarily consistent with the joint posterior peak in the case of underdetermination. Our study theoretically and numerically confirmed that β-dependent SIMMs provide a useful diagnostic tool for the underdetermined mixing problem. In addition to ordinary SIMMs, we recommend reporting the results of β-dependent SIMMs to obtain a biologically feasible and sound interpretation from stable isotope data.

Scientific realism and underdetermination in quantum theory

AbstractThis paper surveys the status of scientific realism in relation to quantum physics, focusing on the problem of underdetermination.

Particle reconstruction of volumetric particle image velocimetry with the strategy of machine learning

Three-dimensional particle reconstruction with limited two-dimensional projections is an under-determined inverse problem that the exact solution is often difficult to be obtained. In general, approximate solutions can be obtained by iterative optimization methods. In the current work, a practical particle reconstruction method based on a convolutional neural network (CNN) with geometry-informed features is proposed. The proposed technique can refine the particle reconstruction from a very coarse initial guess of particle distribution that is generated by any traditional algebraic reconstruction technique (ART) based methods. Compared with available ART-based algorithms, the novel technique makes significant improvements in terms of reconstruction quality, robustness to noise, and at least an order of magnitude faster in the offline stage.

Observations noise and stability of geodetic variable slip fault models

SUMMARY Variable slip models of seismic faults represent ill-posed (or underdetermined) problems with infinity of solutions; a single solution is typically obtained using constraints imposed by the L-curve theory, through an experimentally derived coefficient which optimizes the trade-off between fault roughness (smoothness, mean slip gradient) and mean misfit of observations to the fault model. However, in some cases, diverse variable slip fault models (VSFM) have been presented, even using the same data sets. In this paper, we investigate the problem of stability (sensitivity) of VSFM to input geodetic (GNSS/GPS, INSAR) ground slip measurements, based on statistics of synthetic data: statistically similar sets of synthetic ground displacements were the only degree of freedom in an analysis leading to comparable VSFM, regarded as images of slip amplitude and of rake. Differences of synthetic models from the reference fault model for different levels of noise, both for filtered and for PCA-filtered were examined, and results were summarized in images depicting the differences and the variability (standard deviation) of each fault slip patch, both for slip and rake. From this ‘truth’-based approach, it was found that an increase in observations noise is reflected in increase of variability (instability) of fault models, especially away from the fault centre, higher for slip than in rake. Analysis with a different virtual observations system indicates that stations above the fault tend to show spurious stress concentration areas on the fault surface, in agreement with previous studies. Analysis with synthetic data appears as a promising strategy to validate a VSFM based on geodetic data.

De-combination of belief function based on optimization

In the theory of belief functions, the evidence combination is a kind of decision-level information fusion. Given two or more Basic Belief Assignments (BBAs) originated from different information sources, the combination rule is used to combine them to expect a better decision result. When only a combined BBA is given and original BBAs are discarded, if one wants to analyze the difference between the information sources, evidence de-combination is needed to determine the original BBAs. Evidence de-combination can be considered as the inverse process of the information fusion. This paper focuses on such a defusion of information in the theory of belief functions. It is an under-determined problem if only the combined BBA is available. In this paper, two optimization-based approaches are proposed to de-combine a given BBA according to the criteria of divergence maximization and information maximization, respectively. The new proposed approaches can be used for two or more information sources. Some numerical examples and an example of application are provided to illustrate and validate our approaches.

Deep learning based image reconstruction for sparse-view diffuse optical tomography

Three critical problems, including high-cost instrumentations, time-consuming image recovery, and low image quality, limit clinical applications of diffuse optical tomography (DOT). Image reconstruction based on deep learning can enhance the image quality of DOT, especially in the case of low number of measurements where image reconstruction becomes a more ill-posed and underdetermined problem. Here, we present a sparse-view image reconstruction based on deep learning to recover the absorption coefficient of a phantom. The presented neural network enhances image quality and the results show that deep learning can enhance the contrast to noise-ratio effectively (more than 80%). The presented method provides recovered images with more accurate localization capabilities (an increase in localization metric by more than 30% compared with the classic method). Moreover, the time of image reconstruction is reduced by three orders of magnitude. This method can be applied for single-source DOT that significantly reduces the cost of required instruments for brain imaging and cervical or thyroid cancer screening.

A doubly relaxed minimal-norm Gauss–Newton method for underdetermined nonlinear least-squares problems

When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this type. In this paper, we are concerned with the computation of the minimal-norm solution of an underdetermined nonlinear least-squares problem. We present a Gauss–Newton type method, which relies on two relaxation parameters to ensure convergence, and which incorporates a procedure to dynamically estimate the two parameters, as well as the rank of the Jacobian matrix, along the iterations. Numerical results are presented.

Machine learning model with physical constraints for diffuse optical tomography.

A machine learning model with physical constraints (ML-PC) is introduced to perform diffuse optical tomography (DOT) reconstruction. DOT reconstruction is an ill-posed and under-determined problem, and its quality suffers by model mismatches, complex boundary conditions, tissue-probe contact, noise etc. Here, for the first time, we combine ultrasound-guided DOT with ML to facilitate DOT reconstruction. Our method has two key components: (i) a neural network based on auto-encoder is adopted for DOT reconstruction, and (ii) physical constraints are implemented to achieve accurate reconstruction. Both qualitative and quantitative results demonstrate that the accuracy of the proposed method surpasses that of existing models. In a phantom study, compared with the Born conjugate gradient descent (Born-CGD) reconstruction method, the ML-PC method decreases the mean percentage error of the reconstructed maximum absorption coefficient from 16.41% to 13.4% for high contrast phantoms and from 23.42% to 9.06% for low contrast phantoms, with improved depth distribution of the target absorption maps. In a clinical study, better contrast was obtained between malignant and benign breast lesions, with the ratio of the medians of the maximum absorption coefficient improved from 1.63 to 2.22.

Operational modal identification based on parallel factor decomposition with the presence of harmonic excitation

One of the main difficulties of the operational modal analysis is to deal with underdetermined problems inwhich the number of sensors is less than the number of active modes. In the last decade, methods based on the PARAllel FACtor (PARAFAC) decomposition have attracted a lot of attention in the field of modal analysis because it has been proven that these methods can deal with underdetermined cases, as well as the presence of harmonic excitations. Moreover, in combination with kurtosis value as a harmonic indicator, this makes them more efficient in distinguishing between harmonic and structural components. However, it can lead to distorted results as it does not take into account the variation in the length of the covariance functions of the modal coordinates. Since the kurtosis values are estimated from these covariance functions, the length of the latter directly affects the kurtosis. To overcome this limit, the present study proposes to introduce the choice of the length of these functions based on their frequency and damping coefficient. This change improves the existing method by more efficient separating between harmonics and modal components. The proposed procedure is validated using numerical simulations, followed by ambient vibration measurements. © 2021 Academie des sciences, Paris and the authors.

Leakage detection in water distribution networks by the use of analytical and experimental models

Abstract Recently, calibration based methods are considered to determine the quantities and locations of all leakages in a water distribution network (WDN), simultaneously. In this paper, the accuracy of detecting leakages using Ant Colony Optimization (ACO) in two networks including a hypothetical and a laboratory network is investigated. A novel method is introduced to analyze the effects of measurement errors on demand calibration in an under-determined problem. The results confirm the ability of the optimization-based method in detecting the exact locations and values of the leakages in a WDN. Also, it is emphasized that merely suitable fitnesses cannot provide sufficient confidence in result accuracy. To overcome this difficulty, the correctness of leakage detection can be verified in a sampling design problem. Moreover, to check the reliability of leakage detection an alternative method based on various results is introduced in this research.

Creativity in students\u2019 modelling competencies: conceptualisation and measurement

Modelling competencies are currently included in numerous curricula worldwide and are generally accepted as a complex, process-oriented construct. Therefore, effective measurement should include multiple dimensions, like the sub-competencies required throughout the modelling process. Departing from the characteristics of modelling problems as open and often underdetermined real-world problems, we propose to enrich the current conceptualisation of mathematical modelling competencies by including creativity, which plays an important role in numerous phases of the mathematical modelling process but has scarcely been considered in modelling discourse. In the study described in this paper, a new instrument for the evaluation of this enriched construct has been developed and implemented. The modelling competencies incorporating creativity of the students were evaluated based on the adequacy of the models and the modelling processes proposed, and the appropriateness and completeness of the approaches were evaluated in detail. Adapting measurement approaches for creativity that have been developed in the problem-solving discourse, certain criteria of creativity were selected to evaluate the creativity of the students’ approaches in tackling modelling problems—namely, usefulness, fluency, and originality. The empirical study was conducted among 107 Chinese students at the upper secondary school level, who attended a modelling camp and independently solved three complex modelling problems. The results reveal significant correlations between fluency and originality in students’ performances across all tasks; however, the relationships between usefulness and the other two creativity aspects were not consistent. Overall, the results of the study support the importance of the inclusion of creativity in the construct of modelling competencies.

Technical Note: Temporal disaggregation of spatial rainfall fields with generative adversarial networks

Abstract. Creating spatially coherent rainfall patterns with high temporal resolution from data with lower temporal resolution is necessary in many geoscientific applications. From a statistical perspective, this presents a high- dimensional, highly underdetermined problem. Recent advances in machine learning provide methods for learning such probability distributions. We test the usage of generative adversarial networks (GANs) for estimating the full probability distribution of spatial rainfall patterns with high temporal resolution, conditioned on a field of lower temporal resolution. The GAN is trained on rainfall radar data with hourly resolution. Given a new field of daily precipitation sums, it can sample scenarios of spatiotemporal patterns with sub-daily resolution. While the generated patterns do not perfectly reproduce the statistics of observations, they are visually hardly distinguishable from real patterns. Limitations that we found are that providing additional input (such as geographical information) to the GAN surprisingly leads to worse results, showing that it is not trivial to increase the amount of used input information. Additionally, while in principle the GAN should learn the probability distribution in itself, we still needed expert judgment to determine at which point the training should stop, because longer training leads to worse results.

A new underdetermined NMF based anti-collision algorithm for RFID systems

Radio Frequency Identification (RFID) has been one of the critical technologies of the Internet of Things (IoT). With the rapid development of the IoT, the RFID systems are required to be more efficient and with high throughput capacity. In the widespread IoT application scenes, the collision problem of the RFID tags has become the increasingly remarkable problem in RFID systems. Traditionally, the anti-collision algorithms of RFID systems are always based on time division multiple access (TDMA). Although the TDMA based anti-collision algorithms are simple and easy to implement, it often misses tags and costs high time. Afterwards, the anti-collision algorithms based on blind source separation (BSS) have been introduced. These BSS based anti-collision algorithms are more efficient and stable, but they are mostly suitable for the determined or overdetermined case, i.e., the number of tags is less than that of the readers in RFID systems. Only a few anti-collision algorithms are taken into account of the underdetermined collision model. Because this underdetermined RFID collision model will give rise to more difficult solution but with very meaningfully practical IoT applications. Therefore, to investigate high quality underdetermined anti-collision algorithm for RFID system plays an important role in improving the efficiency of RFID system, and enable RFID implement more wide applications in future IoT systems. As a motivation, this paper proposes a new anti-collision algorithm for underdetermined RFID mixed system for performance improvement. In this work, the nonnegative matrix factorization (NMF) with minimum correlation and minimum volume constrains, i.e., the new MCV_NMF algorithm is proposed for anti-collision application in underdetermined RFID systems. This algorithm combines the independent principle of the tag signals with the NMF mechanism to achieve performance enhancement. The experimental results and analysis corroborate that this new algorithm can implement the underdetermined collision problem well and enhance the throughput capacity of RFID system.

Grounding grid corrosion detection based on mini-batch gradient descent and greedy method

To ensure the speed, recall, and precision of the algorithm to solve the algorithm’s measurement failure due to the problem of high underdetermination and the deviation of some outgoing lines from accessible nodes, the ideal resistance method is proposed to diagnose the corrosion of the grounding grid. The information extraction algorithm, a two-way heuristic network resistance diagnosis method, is added. A non-linear multi-objective optimization model of the deviation between the measured potential and the real potential is established. This paper puts forward that the grounding network’s fault diagnosis is analogous to a neural network’s training process, which makes full use of the excellent robustness and rapidity of a neural network training method. Simultaneously, the algorithm adds the retraining method combined with the field excavation in training and proves its feasibility through the Grobner basis codimension theory in algebraic geometry. Through result verification, it is found that the new algorithm can avoid the inversion failure caused by the algorithm failure due to the lack of accessible nodes and the abnormal part of input data at the same time as a fast solution.

The Song of the Science Mermaid: A Philosophical Trilogue on the Osteological Paradox

As a modern academic Ulysses, the historical scientist is enticed by numerous plausible scientific theories that can explain the historical data in search of the truth. However, the predicament of her work is to inevitably crash onto the rocks and cliffs of uncertainty. The problem discussed in this paper is that several scientific models can be suitable to account for the same empirical observations. The risk of falling into speculation is looming, and exceedingly dangerous in science. This is also the case in archaeological sciences, such as bioarchaeology. A bioarchaeologist frequently encounters traces of disease in ancient skeletons, and pertinent patterns may often result from equally probable different causes. This is a methodological issue commonly encountered in the interpretation of pathological patterns in human remains, and constitutes part of the problem known in bioarchaeology as the osteological paradox. During an informal trilogue, three characters discuss the osteological paradox, and attempt to define it in philosophical terms. The aim of this work is to present the problems of scientists with the philosophical approach to the debate between scientific realism and antirealism, focusing in particular on the so-called problem of underdetermination. Our original approach is to apply the distinction between ‘how-possibly’ models and ‘how-actually’ models by Alisa Bokulich to archaeological issues, integrating various fields of science with a multidisciplinary and omnivorous approach. The trilogue ends providing the historical scientist with reasons and means to believe in her ability to conceive of true and reliable scientific models to interpret the historical past.

Impact force reconstruction and localization using nonconvex overlapping group sparsity

Although extensively studied, impact force identification is still a challenging task. When the location of the impact force is unknown, an under-determined problem is usually required to be tackled. In this paper, a novel impact force identification method based on the nonconvex overlapping group sparsity(NOGS) is proposed, allowing to localize the impact and recover its time history simultaneously from quite limited measurements(i.e., the number of responses is less than the number of potential impact locations). The NOGS not only enriches the prior information by taking the group sparsity structure of impact forces into consideration, but enhances the sparsity and the accuracy of estimated amplitude via its nonconvexity. A new algorithm, named fast nonconvex overlapping group sparsity algorithm(FaNogSa), derived in the light of the Majorize-Minimization(MM) principle is utilized to minimize the nonconvex objective function. Simulations and experiments are both implemented systematically on a stiffened composite structure to validate the proposed method, and two strain gauges are utilized to monitor 54 potential impacts. The corresponding results, comparing to the plain nonconvex(atan) regularization and the standard ℓ1-norm regularization, say that the proposed method is able to localize the impact and at the same time recover its time history accurately, while under the same measuring conditions the nonconvex(atan) method and the ℓ1-norm method usually fail.