Described herein is a framework for analyzing data in high-dimensional space. In accordance with one implementation, observed data and at least one input model parameter set is received. The input model parameter set serves as a solution candidate of a predefined problem (e.g., inverse or optimization problem) and is related to the observed data via a model. To provide enhanced computational efficiency, a reduced base with lower dimensionality is determined based on the input model parameter set. The reduced base is associated with a set of coefficients, which represents the coordinates of any model parameter set in the reduced base. Sampling is performed within the reduced base to generate an output model parameter set in the reduced base. The output model parameter set is compatible with the input model parameter set and fits the observed data, via the model, within a predetermined threshold.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method of determining uncertainty in high-dimensional space, comprising: (i) receiving, by a processor, observed data from a data source and at least one input model parameter set serving as a solution candidate of a predefined problem, wherein the input model parameter set is related to the observed data via a model; (ii) determining based on the input model parameter set, by the processor, a reduced base associated with a set of coefficients that represent coordinates of any model parameter set in the reduced base, wherein the coefficients in the reduced base are fewer than model parameters in the input model parameter set; and (iii) sampling within the reduced base, by the processor, to generate an output model parameter set in the reduced base, wherein the output model parameter set is compatible with the input model parameter set and fits the observed data, via the model, within a predetermined threshold; and (iv) generating, by the processor, one or more uncertainty measures based on the output model parameter set. 2. The method of claim 1 wherein the reduced base comprises a target oriented base. 3. The method of claim 2 wherein the step (ii) comprises determining the target oriented base by applying a learning, interpolation, or conditional simulation technique on one or more secondary attributes, hard model information or a combination thereof. 4. The method of claim 1 wherein the input model parameter set comprises a unique posterior model parameter set. 5. The method of claim 4 wherein the step (ii) comprises orthogonally transforming the posterior model parameter set to determine a posterior reduced base. 6. The method of claim 5 wherein the orthogonally transforming the posterior model parameter set comprises: generating one or more orthogonal transformation matrices by performing a model reduction method; projecting the posterior model parameter set onto at least one of the one or more orthogonal transformation matrices to generate one or more compressed terms; and projecting the one or more compressed terms back onto an original canonical base associated with the posterior model parameter set to generate the posterior reduced base. 7. The method of claim 6 further comprises performing, on the one or more compressed terms, exact compression, group compression, or any combination thereof. 8. The method of claim 6 wherein the model reduction method comprises singular value decomposition, discrete cosine transform, discrete wavelet transform, discrete Chebyshev transform or independent component analysis. 9. The method of claim 5 wherein the posterior reduced base comprises a spatial indicator base. 10. The method of claim 1 wherein the step (iii) comprises hyperprism sampling, minimum norm geometric sampling, telescopic geometric sampling or logarithmic geometric sampling. 11. The method of claim 1 wherein the step (iii) comprises oriented sampling. 12. The method of claim 1 wherein the step (iii) comprises stochastic sampling. 13. The method of claim 12 wherein the step (ii) comprises: receiving prior information; determining a prior reduced base using the prior information; performing optimization in the prior reduced base to locate a low misfit region; and determining a posterior reduced base based on a good model of the low misfit region. 14. The method of claim 13 wherein the optimization comprises particle swarm optimization, differential evolution, Monte Carlo methods, or any combination thereof. 15. The method of claim 1 wherein the one or more uncertainty measures comprise an indicator map, a percentile map, an E-type value, or a combination thereof. 16. The method of claim 1 wherein the predefined problem comprises an inverse problem or an optimization problem. 17. The method of claim 1 wherein the step (iii) further comprises iteratively performing local optimization and hyperquadric sampling in the reduced base. 18. A non-transitory computer readable medium embodying a program of instructions executable by a machine to perform steps for analyzing high-dimensional digital image data, the steps comprising: (i) receiving observed data and a first set of one or more model parameters serving as solution candidates of a predefined problem, wherein said one or more parameters are related to the observed data via a model; (ii) determining, based on the first set of model parameters, a reduced base defined by a second set of model parameters, wherein the second set includes fewer model parameters than the first set; and (iii) sampling within the reduced base to generate a third set of model parameters that is compatible with the first set of model parameters and fits the observed data, via the model, within a predetermined threshold. 19. A system for high-dimensional data analysis, comprising: a memory device for storing computer readable program code; and a processor in communication with the memory device, the processor being operative with the computer readable program code to: (i) receive observed data and a first set of one or more model parameters serving as solution candidates of a predefined problem, wherein said one or more parameters are related to the observed data via a model; (ii) determine, based on the first set of model parameters, a reduced base defined by a second set of model parameters, wherein the second set includes fewer model parameters than the first set; and (iii) sample within the reduced base to generate a third set of model parameters that is compatible with the first set of model parameters and fits the observed data, via the model, within a predetermined threshold. 20. The system of claim 19 wherein the observed data comprises geophysical data, historical production data, pressure data, meteorological data, electrophysiological data, biomedical data or financial data.