Patent Publication Number: US-11379726-B2

Title: Finite rank deep kernel learning for robust time series forecasting and regression

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This Applications claims the benefit of U.S. Provisional Application No. 62/755,229, filed on Nov. 2, 2018, the contents of which are incorporated herein in their entirety. 
    
    
     INTRODUCTION 
     Aspects of the present disclosure relate to computationally efficient methods for forecasting values with confidence intervals based on datasets with complex geometries (e.g., time series data). 
     Forecasting with simultaneous quantification of uncertainty in the forecast has emerged as a problem of practical importance for many application domains, such as: computer vision, time series forecasting, natural language processing, classification, and regression, to name a few. Recent research has focused on deep learning techniques as one possible approach to provide suitable forecasting models, and several approaches have been studied to characterize uncertainty in the deep learning framework, including: dropout, Bayesian neural networks, ensemble-based models, calibration-based models, neural processes, and deep kernel learning. Of these various approaches, deep kernel learning has emerged as a useful framework to forecast values and characterize uncertainty (alternatively, confidence) in the forecasted values simultaneously. In particular, deep kernel learning has proven useful for forecasting time series datasets with complex geometries. 
     Deep kernel learning combines deep neural network techniques with Gaussian process. In this way, deep kernel learning combines the capacity of approximating complex functions with deep neural network techniques with the flexible uncertainty estimation framework of Gaussian process. 
     Unfortunately, deep kernel learning is computationally expensive—generally O(n 3 ), where n is the number of training data points. Thus, when applied to organizations&#39; ever larger and more complex datasets, deep kernel learning may require significant amounts of time and processing resources to operate. And as datasets get larger, the problem grows significantly non-linearly. Consequently, organizations are forced to invest significant resource in additional and more powerful on-site computing resources and/or to offload the processing to cloud-based resources, which are expensive and which may create security concerns for certain types of data (e.g., financial data, personally identifiable data, health data, etc.). 
     Accordingly, what is needed is a framework for reducing the computational complexity of deep kernel learning while still being able to forecast and characterize uncertainty simultaneously. 
     BRIEF SUMMARY 
     Certain embodiments provide a method for performing finite rank deep kernel learning, including: receiving a training dataset; forming a set of embeddings by subjecting the training data set to a deep neural network; forming, from the set of embeddings, a plurality of dot kernels; combining the plurality of dot kernels to form a composite kernel for a Gaussian process; receiving live data from an application; and predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel. 
     Other embodiments comprise systems configured to perform finite rank deep kernel learning as well as non-transitory computer-readable storage mediums comprising instructions for performing finite rank deep kernel learning. 
     The following description and the related drawings set forth in detail certain illustrative features of one or more embodiments. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The appended figures depict certain aspects of the one or more embodiments and are therefore not to be considered limiting of the scope of this disclosure. 
         FIG. 1  depicts an example of a hierarchy of kernel spaces. 
         FIG. 2  depicts a synthetic dataset that has been forecasted using a finite rank deep kernel learning method 
         FIG. 3  depicts finite rank orthogonal embeddings corresponding to the dataset depicted in  FIG. 2 . 
         FIGS. 4A-4D  depicts a first example simulation comparing the performance of different modeling techniques, including deep kernel learning. 
         FIGS. 5A-5D  depicts a second example simulation comparing the performance of different modeling techniques, including deep kernel learning. 
         FIG. 6  depicts a third example simulation comparing the performance of different modeling techniques, including deep kernel learning. 
         FIG. 7  depicts an example of a finite rank deep kernel learning flow. 
         FIG. 8  depicts an example of using a deep neural network to create embeddings for a composite kernel. 
         FIG. 9  depicts an example of a finite rank deep kernel learning method. 
         FIGS. 10A and 10B  depict example application input and output based on a finite rank deep kernel learning model. 
         FIG. 11  depicts an example processing system for performing finite rank deep kernel learning. 
     
    
    
     To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the drawings. It is contemplated that elements and features of one embodiment may be beneficially incorporated in other embodiments without further recitation. 
     DETAILED DESCRIPTION 
     Aspects of the present disclosure provide apparatuses, methods, processing systems, and computer readable mediums for computationally efficiently forecasting values with confidence intervals, which are particularly well-suited to operate on datasets with complex geometries. 
     Deep kernel learning is a state-of-the-art method to forecast with uncertainty bounds (or confidence intervals) that relies upon two underlying machine learning paradigms, namely: deep neural networks and Gaussian process. In deep kernel learning, the deep neural network is used to learn a kernel operator of a Gaussian process, which is then successively used to forecast with uncertainty bounds. 
     Described herein is a modelling framework, which may be referred to a finite rank deep kernel learning, which beneficially reduces the computational complexity of deep kernel learning while enhancing deep kernel learning&#39;s ability to approximate complex functions and estimate uncertainty (or confidence). Notably, while described in the example context of deep kernel learning throughout, the framework described herein is similarly applicable to other kernel-based techniques where deep neural networks are used to learn the kernel, such as: deep kernel learning for classification and deep neural network-based support vector machines, to name a few. 
     One feature of finite rank deep kernel learning is a composite kernel (or “expressive” kernel), which is a linear combination of a plurality of simpler linear (or “dot”) kernels. Composite kernels are capable of capturing complex geometries of a dataset, such as where certain regions of the dataset have very different structure as compared to other regions, even verging on discontinuity. Modeling this type of dataset is difficult with conventional machine learning algorithms, such as deep kernel learning, because traditional machine learning algorithms try to learn a global description of a dataset. 
     In finite rank deep kernel learning, each dot kernel is learned by a deep neural network. Learning simpler dot kernels, which can then be linearly combined into a composite kernel, is easier for the deep neural network to learn because an individual dot kernel represents the local geometry of the dataset rather than the global geometry, which in many cases is more complicated. Because the dot kernels are easier to learn, the overall performance of finite rank deep kernel learning is improved from a processing efficiency standpoint, and any processing device running finite rank deep kernel learning will enjoy improved performance, such as faster operation, lower processing requirements, and less memory usage, as compared to conventional machine learning methods. 
     Example applications of finite rank deep kernel learning include: regression with confidence bounds, forecasting time series or sequential data (long and short term), and anomaly detection, as a few examples. In regression problems based on a set of dependent variables x, the aim is to predict the value of a response variable y≈f(x)+ε. In a time series forecasting problem, z(t) is either a time series or a sequence. The aim is to forecast z(T+τ) based on {a,b(t),z(t)|t=0, , , , T}, where a is the time series metadata and b(t) is the exogenous variable. The time series forecasting problem can be formulated as a regression problem where x={a,b(t),z(t)|t=0, . . . , T}, and y=z(T+τ) in the framework described herein. In both cases, the output of the model would be a probability distribution of the variable y. 
     Gaussian Process Overview 
     A Gaussian process is a stochastic process (e.g., a collection of random variables indexed by time or space) in which every finite collection of those random variables has a multivariate normal distribution. In other words, every finite linear combination of those variables is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. 
     A machine-learning algorithm that involves a Gaussian process uses learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also includes uncertainty information because it is a one-dimensional Gaussian distribution (which is the marginal distribution at that point). Gaussian process is a maximum a posteriori (MAP) framework, which assumes a prior probability distribution over the function space of all possible candidate regressors, and a kernel. Thus, Gaussian process is a flexible and non-parametric framework that can forecast and quantify uncertainty simultaneously. 
     Generally speaking, there are two approaches to derive the theory of the Gaussian process: weight space view and function space view. 
     In the weight space view approach, the weights of a regression are assumed to be derived from a probability distribution, which has a Gaussian prior. A maximum a posteriori estimation is utilized to evaluate the posterior, which is then used to weigh predictions for test points for different weight configurations. 
     In the function space view approach, the candidate regressor functions are assumed to be sampled from a probability distribution of functions. The kernel operator models the covariance, and the posterior distribution is used to average the predictions made by individual regressors. The following outlines the equations for the function space viewpoint. 
     Initially, let X={x 1 , . . . x n } be the training features, where x i ∈   d , and f(X):={f(x 1 ), . . . f(x n )} is sampled from a distribution  (0, K X,X ), where K X,X ∈   n×n  is comprised of the value of the kernel operator evaluated at every pair of training data points. Further, let y denote the response variable corresponding to the training data points. It can be shown that:
 
 f   *   |X   *   ,X,y,γ,σ   2 ˜ ( E [ f   * ], cov[ f   * ]),
 
 E [ f   * ]= K   X     *     ,X ( K   X,X +σ 2   I ) −1   y,  
 
Cov[ f   * ]= K   X     *     ,X     *     −K   X     *     ,X [ K   X,X +σ 2   I ] −1   y.  
 
     Selection of the kernel function in Gaussian process is a non-trivial task, as depending on the geometry of the data, the right similarity or kernel function needs to be identified. 
     A loss function may be constructed as the log likelihood on the posterior distribution of the Gaussian process, and during the model training, the error is successively back-propagated to find the optimal embedding and the radial basis function scale parameter. 
     Notably, the algorithmic complexity of Gaussian process is O(n 3 ), where n is the number of training data points. In order to reduce the computational complexity, a framework is described herein to derive the kernel with a complexity of O(n 2 ) without any approximation of the kernel. 
     Deep Kernel Learning Overview 
     In the deep kernel learning framework, a deep neural network is used to learn an embedding, which is acted upon by a radial basis function (RBF) to create the kernel. A radial basis function (RBF) is a real-valued function ϕ whose value depends only on the distance from the origin, so that ϕ(x)=ϕ(∥x∥); or alternatively on the distance from some other point c, called a center, so that ϕ(x,c)=ϕ(∥x−c∥). Sums of radial basis functions may be used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network. 
     The imposition of a radial basis function kernel adds an additional structure to the kernel operator, which is a continuous operator. However, such kernels may not be adequate to represent the geometry of an arbitrary dataset, especially in cases where the dataset has widely varying local geometries. Thus, as described further herein, the kernel may be created instead as a linear combination of a set of simpler dot kernels. The decomposition of the kernel in this manner (i.e., using a plurality of dot kernels) ensures that each kernel captures the local geometry of a portion of the dataset and the linearly combined (composite) kernel captures the combined geometry of the whole dataset. As described further below, this may be achieved by constructing orthogonal embeddings as deep neural network outputs. This approach allows simultaneous unsupervised learning (e.g., clustering) and supervised learning (e.g., regression) in a unified framework. 
     The Gaussian process kernel may be defined as k(x i ,x j ) and a kernel of deep kernel learning may be defined as:
 
 k ( x   i   ,x   j )→ k (ϕ( x   i ,ω),ϕ( x   j ,ω)|ω,θ)
 
where, ϕ(x i ,ω) is the embedding learnt by the deep neural network. The set of parameters are defined as γ=[ω,θ]. The log marginal likelihood function   is minimized for the Gaussian process, and back-propagation is used to optimize simultaneously ω and θ, which gives the following equations:
 
     
       
         
           
             
               
                 
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     Notably, the complexity of computing K γ   −1  is  (n 3 ). By contrast, in finite rank deep kernel learning, a deep neural network is configured to learn the embeddings as mutually orthogonal, which is successively used to reduce the computational complexity in computation of K γ   −1  to  (n 2 ). 
     Two challenges of conventional deep kernel learning are reducing computational cost without approximating the kernel function and the representation power of the kernel function. The finite rank deep kernel learning framework described herein enhances the representation power, while at the same time reducing the computational complexity. In other words, the framework described herein improves the performance of any machine upon which it is running (through reduced computation complexity) as well as improves the performance of whatever application it is supporting (through improved representation power). 
     Hierarchy of Kernel Operators 
     Let X be an arbitrary topological space, which is a feature space of a regression. Let H be the Hilbert space of the bounded real valued functions defined on X. Initially, a kernel operator K:X×X→  is called positive definite when the following is true:
 
 K ( x,y )= K ( y,x )
 
Σ i,j=1   n   c   i   c   j   K ( x   i   ,x   j )≥0,∀ n∈     ,x   i   ,x   j   ∈X,c   i   ∈R.  
 
     With the aid of the Riesz Representation Theorem, it can be shown that for all x∈X, there exists an element K x ∈H, such that f(x)= f,L x   , where  ·,·  is an inner product, with which the Hilbert Space H is endowed. Next, a reproducing kernel for the Hilbert space H (RKHS) may be defined, which constructs an operator K(x,y) as an inner product of two elements K x  and K y  from the Hilbert space H. A reproducing kernel for the Hilbert space H may be defined as:
 
 K ( x,y ):=   K   x   ,K   y     ,∀x,y∈X.  
 
     From the definition of the reproducing kernel for the Hilbert space H, it can be observed that the reproducing kernel for the Hilbert space H satisfies the conditions of the positive definite kernel operators, as described above. Moreover, the Moore Aronszajn Theorem proves that for any symmetric positive definite kernel operator K, there exists a Hilbert space H for which it is the reproducing kernel for the Hilbert space H, or in other words, the operator satisfies K(x,y):= K x ,K y   , where K x  and K y  belongs to the Hilbert space H of the real valued bounded functions on X. Notably, K x  and K y  can be discontinuous. 
     It also can be noted that the space of the reproducing kernel for the Hilbert space H may be very rich in terms of the complexity, and no a priori assumption need be made on the smoothness of the operator. An example of a reproducing kernel for the Hilbert space H, which is non-smooth is as follows: 
     
       
         
           
             
               
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     Thus, K=δ is a symmetric and positive definite kernel, but is non-smooth. 
     Next, a subclass of the reproducing kernel for the Hilbert space H may be considered, which is continuous in addition to being symmetric and positive definite. Such kernels are called Mercer kernels. Mercer&#39;s Decomposition Theorem provides a decomposition of such an arbitrary kernel into the Eigen functions, which are continuous themselves. For example, for any continuous reproducing kernel for the Hilbert space K(x,y), the following condition is satisfied: 
     
       
         
           
             
               
                 
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     where Θ i ∈   0  forms a set of orthonormal bases, and ζ i ∈   +  are the i th  Eigen function and Eigen value of the integral operator T k (⋅), corresponding to the kernel K. It also can be shown with the aid of the spectral theorem that the Eigen values asymptotically converge to 0. 
     Kernel operators may be thought of as similarity functions that capture relationships between points in a dataset.  FIG. 1  depicts an example of a hierarchy of kernel spaces, including rank 1 kernels  102 , finite rank kernels  104 , Mercer kernels  106 , and reproducing kernels for the Hilbert space  108 . 
     Kernel functions used in existing deep kernel learning methods are primarily radial basis function kernels and polynomial kernels and consequently form a small set of possible kernel functions to represent a potentially rich and complex data set. These kernels are primarily rank 1 (e.g.,  102 ), which constitute a smaller subspace of possible kernels as depicted in  FIG. 1 . 
     By contrast, finite rank deep kernel learning expresses a composite kernel as a sum of multiple simpler dot kernels, which cover the space of finite rank Mercer kernels  104 , as depicted in  FIG. 1 . The composite kernels may be expressed as follows:
 
 K ( x,y )=Σ i=1   R ϕ i ( x )·ϕ i ( y ),
 
where ϕ i (x)&#39;s form a set of orthogonal embeddings, which are learnt by a deep neural network. By expressing the composite kernel in this fashion, one can show that the possible set of kernels would become richer than the existing kernels adopted in conventional deep kernel learning approaches.
 
     As depicted in  FIG. 1 , Mercer kernels  106  form a smaller subspace of the reproducing kernels for the Hilbert space  108 , as Mercer kernels are generally continuous. For instance, the kernel δ(x,y), which is a reproducing kernel for the Hilbert space, but is not continuous, cannot be decomposed according to the Mercer&#39;s Decomposition Theorem. But a rich subset of kernels can be represented by Mercer kernels, which can be expressed as follows:
 
 K ( x,y )˜Σ i=1   R ζ i Θ i ( x )Θ i ( y ),
 
     where Θ i  forms an orthonormal basis. The orthonormality of the basis ensures an inverse of the operator can be constructed as follows: 
     
       
         
           
             
               
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     which reduces computation of the inverse operator. Further, the Mercer kernels  106  can have countable ranks with diminishing Eigen values. So, while Mercer kernels  106  are a less rich set as compared to the reproducing kernels for the Hilbert space  108 , they nevertheless form a diverse subspace of the reproducing kernels for the Hilbert space  108 . 
     Notably, a Mercer kernel generally has countable rank i.e. any Mercer kernel can be expressed as sum of countable rank 1 kernels. For example, Θ i (x)Θ i (y) is a rank 1 kernel. Generally, a Mercer kernel K(x,y) is of finite rank R if it can be expressed as follows:
 
 K ( x,y )=Σ i=1   R ζ j Θ i ( x )Θ i ( y ).
 
     In some cases, kernels used in machine learning are rank 1 Mercer kernels, which have σ 1 =1, and σ i =0 for i≥2. For example, popular kernels used in machine learning, such as a polynomial kernel (k(x,y)=(x′y+c) d ) and a Radial Basis Function kernel 
               (       k   ⁡     (     x   ,   y     )       =     exp   ⁡     (     -              x   -   y          2       σ   2         )         )     ,         
are rank 1 Mercer kernels. Generally, any rank 1 Mercer kernel may be expressed as K(x,y)= c(x),c(y)  for some continuous function c.
 
     As above, Mercer kernels  106  form a subspace of the reproducing kernels for the Hilbert space  108 . Similarly, rank 1 Mercer kernels  102  form a smaller subspace of the Mercer kernels  106 , as depicted in  FIG. 1 . 
     Finite Rank Deep Kernel Learning 
     It is desirable to create a set of kernels that have greater representational power. One method is to use a finite rank Mercer kernel to represent a richer class of kernels, which may be represented as follows:
 
 K ( x,y )=Σ i=1   R ϕ i ( x ,ω)ϕ i ( y ,ω)
 
     This kernel selection technique is useful when using deep neural networks to learn the embeddings ϕ i , especially where a dataset has widely differing local geometries, because the deep neural network could decompose the embeddings ϕ i  into orthogonal embeddings. 
     Notably, any arbitrary finite rank Mercer kernel can be approximated by a deep neural network. Thus, for any Mercer kernel:
 
 K ( x,y )=Σ i=1   R ζ i Θ i ( x )Θ i ( y ),
 
and an ε&gt;0,
 
     there exists an N and a family of neural network regressors with finite number of hidden units and output layer ϕ i (z,w) such that: 
     
       
         
           
             
               
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     where Θ i (z,w) forms a set of orthogonal functions. Accordingly, an arbitrary smooth Mercer kernel can be modeled by a multi-layer neural network with outputs Θ i (z,w). The outputs of the neural network may form embeddings that are orthogonal to one another. As a consequence of the orthogonality, the inverse operator can also be expressed in terms of the deep neural network output layer. 
     Generation of Orthogonal Embeddings 
     The Gaussian process kernel may be modeled as follows:
 
 K ( x,y )=Σ i=1   R ϕ i ( x,w )ϕ i ( y,w ),
 
where ϕ i (y,w) would be ideally orthogonal to one another. The deep kernel learning algorithm optimizes the negative log likelihood function, based on the kernel operator, which may be represented as follows:
 
−log  p ( y|x )˜ y   T ( K   γ +σ 2   I ) −1   y +log| K   γ +σ 2   I| 
 
     A penalty term to the cost may be introduced, as follows: 
                 -   log     ⁢           ⁢     p   ⁡     (     y   ❘   x     )         +     λ   ⁢       ∑     i   ,   j   ,     i   ≠   j         ⁢       (           ϕ   i     ⁡     (     x   ,   w     )       T     ⁢       ϕ   j     ⁡     (     x   ,   w     )         )     2               
where, λ is a weight assigned to the orthogonality objective as opposed to the log likelihood. Notably, Σ i,j,i≠j (ϕ i (x,w) T ϕ j (x,w)) 2  is minimized when the embeddings are orthogonal.
 
     Below it will be shown further that the inversion of the matrix (K γ +σ 2 I) −1 , and the determinant computation |K γ +σ 2 I| can be further simplified, and as a consequence the optimization can be done in batches. 
     Computational Complexity 
     The main computational bottleneck in deep kernel learning concerns the inversion of the kernel operator, which as above has computational complexity O(n 3 ). By contrast, expressing the Gaussian process kernel as a linear combination of dot kernels as with finite rank deep kernel learning means that the composite kernel operator can be inverted as follows: 
     
       
         
           
             
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     This construction reduces the complexity to O(n 2 ) i.e., an order of magnitude. This reduction in complexity significantly improves the performance of finite rank deep kernel learning over conventional deep kernel learning. For example, processing systems implementing finite rank deep kernel learning will have to expend fewer processing resources and memory resources, and the processing itself will complete significantly quicker, thereby improving the performance of the processing machine as well as the application being supported by the processing machine. 
     The following equations demonstrate the reduction in computational complexity in more detail. As above, the kernel operator may be expressed as follows:
 
 K ( x,y )=Σ i=1   R ϕ i ( x,w )ϕ i ( y,w ),
 
     where the embeddings ϕ i (y,w) are orthogonal to each other. For a finite number of training data points, the embeddings may be defined as follows:
 
Φ i =[ϕ i (( x   1   ,w )), . . . ϕ i (( x   n   ,w ))]
 
     The kernel matrix is thus defined as:
 
 K=Σ   i=1   R Φ i ·Φ i   T .
 
     The orthogonality is also useful in terms of reducing the computational complexity, while inverting the kernel operator (as discussed above): 
     
       
         
           
             
               
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     It can be shown using the orthogonality property of the embeddings that: 
     
       
         
           
             
               
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                     y 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       R 
                     
                     ⁢ 
                     
                       
                         Φ 
                         i 
                       
                       · 
                       
                         Φ 
                         i 
                         T 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   ( 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       R 
                     
                     ⁢ 
                     
                       
                         
                           1 
                           
                             
                                
                               
                                 Φ 
                                 i 
                               
                                
                             
                             2 
                           
                         
                         · 
                         
                           Φ 
                           i 
                         
                       
                       ⁢ 
                       
                         Φ 
                         i 
                         T 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 I 
                 . 
               
             
           
         
       
     
     Next, an expression may be derived for (K+σ 2 I) −1  from the kernel operator. In particular, an identity matrix may be expressed as: 
     
       
         
           
             I 
             = 
             
               ( 
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     R 
                   
                   ⁢ 
                   
                     
                       
                         1 
                         
                           
                              
                             
                               Φ 
                               i 
                             
                              
                           
                           2 
                         
                       
                       · 
                       
                         Φ 
                         i 
                       
                     
                     ⁢ 
                     
                       Φ 
                       i 
                       T 
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       1 
                     
                     
                       n 
                       - 
                       R 
                     
                   
                   ⁢ 
                   
                     
                       
                         1 
                         
                           
                              
                             
                               η 
                               j 
                             
                              
                           
                           2 
                         
                       
                       · 
                       
                         η 
                         j 
                       
                     
                     ⁢ 
                     
                       η 
                       j 
                       T 
                     
                   
                 
               
               ) 
             
           
         
       
     
     where η j &#39;s are orthogonal to each other and form the null space of the subspace spanned by the embeddings Φ i . This construction leads to the following equations, which provides the formulae to invert matrices, which may be used for training and inference with reduced computational complexity: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 ( 
                 
                   K 
                   + 
                   
                     
                       σ 
                       2 
                     
                     ⁢ 
                     I 
                   
                 
                 ) 
               
               = 
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     R 
                   
                   ⁢ 
                   
                     
                       
                         
                           σ 
                           2 
                         
                         
                           
                              
                             
                               Φ 
                               i 
                             
                              
                           
                           2 
                         
                       
                       · 
                       
                         Φ 
                         i 
                       
                     
                     ⁢ 
                     
                       Φ 
                       i 
                       T 
                     
                   
                 
                 + 
                 
                   
                     σ 
                     2 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       
                         n 
                         - 
                         R 
                       
                     
                     ⁢ 
                     
                       
                         
                           1 
                           
                             
                                
                               
                                 η 
                                 j 
                               
                                
                             
                             2 
                           
                         
                         · 
                         
                           η 
                           j 
                         
                       
                       ⁢ 
                       
                         η 
                         j 
                         T 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   K 
                   * 
                 
                 ⁡ 
                 
                   ( 
                   
                     K 
                     + 
                     
                       
                         σ 
                         2 
                       
                       ⁢ 
                       I 
                     
                   
                   ) 
                 
               
               
                 - 
                 1 
               
             
             = 
             
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     R 
                   
                   ⁢ 
                   
                     
                       
                         
                           σ 
                           2 
                         
                         
                           
                              
                             
                               Φ 
                               i 
                             
                              
                           
                           2 
                         
                       
                       · 
                       
                         Φ 
                         i 
                       
                     
                     ⁢ 
                     
                       Φ 
                       i 
                       T 
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       1 
                     
                     
                       n 
                       - 
                       R 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           σ 
                           2 
                         
                         
                           
                              
                             
                               η 
                               j 
                             
                              
                           
                           2 
                         
                       
                       · 
                       
                         η 
                         j 
                       
                     
                     ⁢ 
                     
                       η 
                       j 
                       T 
                     
                   
                 
               
               = 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       R 
                     
                     ⁢ 
                     
                       
                         Φ 
                         i 
                         * 
                       
                       · 
                       
                         
                           ( 
                           
                             Φ 
                             i 
                             * 
                           
                           ) 
                         
                         T 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         R 
                       
                       ⁢ 
                       
                         
                           
                             
                               σ 
                               2 
                             
                             
                               
                                  
                                 
                                   Φ 
                                   i 
                                 
                                  
                               
                               2 
                             
                           
                           · 
                           
                             Φ 
                             i 
                           
                         
                         ⁢ 
                         
                           Φ 
                           i 
                           T 
                         
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     The next step addresses the question of whether or not the loss function can be decomposed over the training data points. The following set of equations leads to the decomposition of the cost function into training data points: 
     
       
         
           
             
               
                 
                   - 
                   log 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   p 
                   ⁡ 
                   
                     ( 
                     
                       y 
                       ❘ 
                       x 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   ∼ 
                 
                 ⁢ 
                 
                   
                     
                       y 
                       T 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           K 
                           γ 
                         
                         + 
                         
                           
                             σ 
                             2 
                           
                           ⁢ 
                           I 
                         
                       
                       ) 
                     
                   
                   
                     - 
                     1 
                   
                 
                 ⁢ 
                 y 
               
               + 
               
                 log 
                 ⁢ 
                 
                    
                   
                     
                       K 
                       γ 
                     
                     + 
                     
                       
                         σ 
                         2 
                       
                       ⁢ 
                       I 
                     
                   
                    
                 
               
             
             = 
             
               
                 
                   
                     
                       
                         y 
                         T 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               R 
                             
                             ⁢ 
                             
                               
                                 Φ 
                                 i 
                               
                               ⁢ 
                               
                                 Φ 
                                 i 
                                 T 
                               
                             
                           
                           + 
                           
                             
                               σ 
                               2 
                             
                             ⁢ 
                             I 
                           
                         
                         ) 
                       
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   y 
                 
                 + 
                 
                   log 
                   ⁢ 
                   
                      
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           R 
                         
                         ⁢ 
                         
                           
                             Φ 
                             i 
                           
                           ⁢ 
                           
                             Φ 
                             i 
                             T 
                           
                         
                       
                       + 
                       
                         
                           σ 
                           2 
                         
                         ⁢ 
                         I 
                       
                     
                      
                   
                 
               
               = 
               
                 
                   
                     
                       
                         
                           y 
                           T 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ∏ 
                               
                                 i 
                                 = 
                                 1 
                               
                               R 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     1 
                                     
                                       σ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     Φ 
                                     i 
                                   
                                   ⁢ 
                                   
                                     Φ 
                                     i 
                                     T 
                                   
                                 
                                 + 
                                 
                                   
                                     σ 
                                     2 
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     y 
                   
                   + 
                   
                     log 
                     ⁢ 
                     
                        
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           R 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 1 
                                 
                                   σ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 Φ 
                                 i 
                               
                               ⁢ 
                               
                                 Φ 
                                 i 
                                 T 
                               
                             
                             + 
                             
                               
                                 σ 
                                 2 
                               
                               ⁢ 
                               I 
                             
                           
                           ) 
                         
                       
                        
                     
                   
                 
                 = 
                 
                   
                     
                       
                         y 
                         T 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ∏ 
                             
                               i 
                               = 
                               1 
                             
                             R 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     1 
                                     
                                       σ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     Φ 
                                     i 
                                   
                                   ⁢ 
                                   
                                     Φ 
                                     i 
                                     T 
                                   
                                 
                                 + 
                                 
                                   
                                     σ 
                                     2 
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     y 
                   
                   + 
                   
                     log 
                     ⁢ 
                     
                        
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           R 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 1 
                                 
                                   σ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 Φ 
                                 i 
                               
                               ⁢ 
                               
                                 Φ 
                                 i 
                                 T 
                               
                             
                             + 
                             
                               
                                 σ 
                                 2 
                               
                               ⁢ 
                               I 
                             
                           
                           ) 
                         
                       
                        
                     
                   
                 
               
             
           
         
       
     
     Notably, the preceding formulas are derived assuming orthogonality of the embeddings ϕ i . However, they would be approximately orthogonal since orthogonality has been modeled as a continuous loss function. 
     Deep Orthogonal Bayesian Linear Regression 
     Parallels may be drawn between the Gaussian process and Bayesian linear regression. In fact, Bayesian linear regression can be shown as equivalent to Gaussian process when employing dot kernels, as described above. And similar to Gaussian process, a set of orthogonal embedding functions can be used in Bayesian linear regression as follows: 
     
       
         
           
             
               
                 y 
                 ⁢ 
                 
                   ∼ 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     R 
                   
                   ⁢ 
                   
                     
                       
                         ϕ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ⁢ 
                     
                       w 
                       i 
                     
                   
                 
               
               + 
               ɛ 
             
             , 
           
         
       
     
     where, ε is the observation noise with variance σ 2 , and R is the rank of the combined embedding. Therefore, it can be shown that: 
     
       
         
           
             
               
                 f 
                 * 
               
               ❘ 
               
                 x 
                 * 
               
             
             , 
             X 
             , 
             
               y 
               ⁢ 
               
                 ∼ 
               
               ⁢ 
               
                 𝒩 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         1 
                         
                           σ 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           R 
                         
                         ⁢ 
                         
                           
                             
                               ϕ 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 * 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             A 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             Φ 
                             i 
                           
                           ⁢ 
                           y 
                         
                       
                     
                     , 
                     
                       
                         
                           ϕ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           
                             x 
                             * 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         A 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         ϕ 
                         ⁡ 
                         
                           ( 
                           
                             x 
                             * 
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     where Φ i =Φ i (X), and 
             A   =         ∑     i   =   1     R     ⁢       1     σ   2       ⁢     Φ   i     ⁢     Φ   i   T         +       ∑   p     -   1       .             
This can be further modified as follows:
 
     
       
         
           
             
               
                 f 
                 * 
               
               ❘ 
               
                 x 
                 * 
               
             
             , 
             X 
             , 
             
               y 
               ⁢ 
               
                 ∼ 
               
               ⁢ 
               
                 𝒩 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         R 
                       
                       ⁢ 
                       
                         
                           
                             ϕ 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               * 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             p 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   Φ 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     K 
                                     + 
                                     
                                       
                                         σ 
                                         2 
                                       
                                       ⁢ 
                                       I 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             y 
                           
                         
                       
                     
                     , 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         R 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 ϕ 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   * 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 p 
                               
                               ⁢ 
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     * 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             
                               
                                 ϕ 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   * 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 p 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       Φ 
                                       i 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         K 
                                         + 
                                         
                                           
                                             σ 
                                             2 
                                           
                                           ⁢ 
                                           I 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   Φ 
                                   i 
                                   T 
                                 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     p 
                                   
                                   ⁢ 
                                   
                                     
                                       ϕ 
                                       i 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         * 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     where K=Σ i=1   R  Φ i Σ p Φ i . With orthogonality of the preceding formula, the complexity of (K+σ 2 I) −1  may be reduced as demonstrated above. Thus, a global optimization problem may be formulated, which would optimize the log likelihood of the Bayesian linear regression with orthogonal embeddings, learnt by a deep neural network. And the orthogonality constraint in the loss function may be implemented as described above. 
     Example Simulations of Finite Rank Deep Kernel Learning 
       FIG. 2  depicts a synthetic dataset that has been forecasted using a finite rank deep kernel learning method as described herein. The corresponding finite rank orthogonal embeddings ϕ i (x) (based on the dataset depicted in  FIG. 2 ) are depicted in  FIG. 3 . As depicted, the dataset in  FIG. 2  has a complex geometry wherein different portions  202 ,  204 , and  206  of the dataset have distinctly different geometries. Depicted in  FIG. 2  are training samples, a “true” function, a predicted function (via finite rank deep kernel learning), and a predicted standard deviation, which acts as an uncertainty or confidence interval. 
     As depicted in  FIG. 2 , the predicted function closely approximates the true function in each of the different portions ( 202 ,  204 , and  206 ), despite the very different geometries in those sections. 
       FIG. 3  depicts an example of the approximately orthogonal embeddings ϕ i (x) ( 302 ) that form the basis of the dot kernels, which when combined into a composite (i.e., expressive) kernel, create the output depicted in  FIG. 2 . The orthogonality of the embeddings shows the expressive power of the linear combination of dot kernels (i.e., composite kernel). In particular, these dot kernels tease out the local relationships in the underlying dataset. The dot kernels can further be combined to form the Gaussian process kernel. Hence, the dot kernels inherently identify clusters of data in the dataset while learning the pattern of the overall dataset simultaneously. The clustering of the dataset into groups allows the neural network to learn the local geometries in a decoupled fashion. The benefit of this approach is that the deep neural network can more accurately fit regions of the dataset in cases of datasets with discontinuities. 
       FIGS. 4A-4D  depicts a first example simulation comparing the performance of different modeling techniques, including deep kernel learning depicted in  FIG. 4A , Gaussian process depicted in  FIG. 4B , bag of neural networks depicted in  FIG. 4C , and finite rank deep kernel learning depicted in  FIG. 4D , which is described herein, using a time series dataset. In this example, the time series dataset is based on a sinusoidal function whose frequency increases as the square of x. Additionally, heteroscedastic noise has been added to the function such that the noise magnitude increases from the left to right. 
     It is apparent from the simulation results that the bag of neural network method of  FIG. 4C  underestimates the confidence intervals near the noisy region and overestimates the confidence in the high frequency region. For both deep kernel learning of  FIG. 4A  and Gaussian process of  FIG. 4B , it is apparent that the confidence intervals fluctuate heavily near the noisy region. 
     By contrast, the finite rank deep kernel learning method of  FIG. 4D  produces confidence bounds that are relatively stable and that capture regions of high noise and fluctuations. Thus, the finite rank deep kernel learning method shows significant improvement in the ability to both forecast values based on a data set with complex geometry while providing accurate quantification of uncertainty via confidence intervals as compared to the conventional methods depicted in  FIGS. 4A-4C . 
       FIGS. 5A-5D  depicts simulation results for a logistic map example. In this example, the logistic map is a chaotic but deterministic dynamical system x n+1 =rx n (1−x n ), where x n ∈S 1 . The time series data is generated by the system for r=4.1, which falls in the region of strange attractor. In general, strange attractor signifies deterministic chaos, which is difficult to forecast. Thus, performance of a modeling framework, such as finite rank deep kernel learning, may be assessed by, for example, modeling a time series that is deterministically chaotic. 
     In this example, the Gaussian process of  FIG. 5B  and bag of neural network of  FIG. 5C  have outputs with overly wide confidence intervals, and these models erroneously identify chaos as noise. The deep kernel learning output of  FIG. 5A  has confidence bounds that are relatively moderate, but does not track the true function particularly closely. 
     By contrast, the finite rank deep kernel learning method of  FIG. 5D  correctly captures the chaotic time series with very narrow confidence bounds. 
       FIG. 6  is an example of simulation results based on a regression dataset. In this case, a normalized root mean squared error was calculated as a measure of accuracy, which is computed as the root mean squared error of a predictor divided by the standard error of the samples. In general, a normalized root mean squared error&lt;1 would be a threshold for any predictor performing better than the sample mean. 
     In this example, the normalized root mean squared error values were found to be 0.41 for deep kernel learning (a) and 0.25 for finite rank deep kernel learning (d)—representing a near 40% improvement in model predictive performance. Further, the average CPU time lapsed for one epoch during the model training was 0.32 sec for deep kernel learning and 0.079 sec for finite rank deep kernel learning—representing a near 76% improvement. Further, the inference time was 0.03 seconds for deep kernel learning and 0.01 seconds for finite rank deep kernel learning—representing a 67% performance improvement. 
     Taken collectively,  FIGS. 4-6  demonstrate that finite rank deep kernel learning outperforms conventional modeling methods, including deep kernel learning, both in terms of accuracy as well as computational efficiency. 
     Example Finite Rank Deep Kernel Learning Flow 
       FIG. 7 . depicts a finite rank deep kernel learning flow  700 . Flow  700  begins at step  702  with acquiring training data. In some cases, as described above, the training data may form a complex geometry. 
     Flow  700  then proceeds to step  704  where the training data is fed to a deep neural network. The outputs of the deep neural network are embeddings. As described above, by changing the loss function to emphasize orthogonality of the embeddings, the resulting embeddings may be approximately orthogonal.  FIG. 8 , described below, depicts an example of using a deep neural network to construct embeddings. 
     Flow  700  then proceeds to step  706  where a plurality of dot kernels are constructed from the approximately orthogonal embeddings. As described above, each of the dot kernels may be a finite rank Mercer kernel.  FIG. 8 , described below, depicts an example of forming a dot kernel from embeddings produced by a deep neural network. 
     Flow  700  then proceeds to step  708  where the plurality of dot kernels are linearly combined into a composite (i.e., expressive) kernel. 
     Flow  700  then proceeds to step  710  where the composite kernel is used as the basis of a finite rank deep kernel learning predictive model. For example, the predictive model may act on live data  712  to create predictions. Notably, the predictive model has two separate outputs, including the predicted values, which are mean values based on the predictions from each dot kernel, and a confidence associated with the predicted values. 
     Flow  700  concludes at step  714  where an application, such as computer vision, time series forecasting, natural language processing, classification, and regression, or those described in more detail below, uses the outputs of the predictive model to provide an application-specific output  716 . 
     Example of Using Deep Neural Network to Create Embeddings for a Composite Kernel 
       FIG. 8  depicts an example of using a deep neural network to create embeddings for a composite kernel for finite rank deep kernel learning. As depicted in  FIG. 8 , a deep neural network with multiple hidden layers  802  learns a plurality of embeddings  804 , which in this example include ϕ 1 (x), ϕ 2 (x), and ϕ 3 (x). The plurality of embeddings are used to form the dot kernel for the Gaussian process  806 , which in this example is rank 3. In this example, the deep neural network produces vector output, each entry of which forms an individual embedding. 
     Example Method for Creating Finite Rank Deep Kernel Learning Model 
       FIG. 9  depicts a finite rank deep kernel learning method  900 . 
     Method  900  begins at step  902  with receiving a training dataset. As described above, in some examples the training dataset may comprise training data with complex geometries, such as datasets including multiple discontinuous portions. 
     Method  900  then proceeds to step  904  with forming a set of embeddings by subjecting the training data set to a deep neural network. 
     Method  900  then proceeds to step  906  with forming, from the set of embeddings, a plurality of dot kernels. In some examples, Σ i,j,i≠j  (ϕ i (x,w) T ϕ j (x,w)) 2  is minimized as a cost function to maintain an orthogonality of the set of embeddings when forming the set of embeddings. 
     In some examples, the orthogonality of the set of embeddings is optimized based on a cost function, wherein the cost function includes a penalty term λ associated with orthogonality of the set of embeddings. For example, the cost function may be implemented as: −log p(y|x)+λΣ i,j,i≠j  (ϕ i (x,w) T ϕ j (x,w)) 2 . 
     Method  900  then proceeds to step  908  with combining the plurality of dot kernels to form a composite kernel. In one example, the composite Kernel for the Gaussian process is modeled as a linear combination of the plurality of dot kernels as: K(x,y)=Σ i=1   R  ϕ i (y,w). In some examples, the composite kernel for the Gaussian process is a finite rank Mercer kernel. 
     Method  900  then proceeds to step  910  with receiving live data from an application. In this example, live data is distinguished from training data in that it is data the model has not yet seen. In one example, the live data comprises financial data. In another example, the live data comprises resource utilization data. In yet another example, the live data is user activity data (e.g., user log data captured on systems used by users or accessed by users). 
     Method  900  then proceeds to step  912  with predicting a plurality of values and a plurality of uncertainties associated with the plurality of values simultaneously using the composite kernel. 
     In one example, the application is a financial management application, the plurality of values comprises a plurality of predicted future financial transactions, and each uncertainty of the plurality of uncertainties associated with a respective predicted future financial transaction estimates a range of values of the respective predicted future transaction. 
     In another example, the application is a resource management application, the plurality of values comprises a plurality of predicted resources needs, and each uncertainty of the plurality of uncertainties associated with a respective predicted future resource need estimates a range of values of the respective resource need. 
     In yet another example, the application is a resource access control application, the plurality of values comprises a plurality of predicted user activities, and each uncertainty of the plurality of uncertainties associated with a respective predicted future user activity estimates a range of values of the respective user activity. 
     Notably,  FIG. 9  is just one example, and other methods including different aspects are described herein. 
     Example Applications for Finite Rank Deep Kernel Learning 
     Finite rank deep kernel learning is broadly applicable to any application where forecasting or prediction with uncertainty estimation (e.g., confidence intervals) is necessary. In particular, finite rank deep kernel learning may improve application and processing system performance where the underlying dataset is large and complex (e.g., in terms of geometry). 
     A first example application is cash flow forecasting e.g., for providing financial services. For example, a financial services organization may seek to provide cash flow forecasting services to users of a financial planning application. Because cash flow, when considered as a time series dataset, tends to be complex and discontinuous, finite rank deep kernel learning (as described herein) is a desirable methodology. Moreover, because the organization offering the financial services may have many customers with much data that can be aggregated, finite rank deep kernel learning may significantly improve the performance of the financial planning application in terms of accuracy of cash flow forecasts, confidence in cash flow forecasts, speed of generating the forecasts, and efficiency of computing resources used to produce the forecasts and uncertainty estimations (e.g., confidence intervals). 
       FIGS. 10A and 10B  depict example application input and output based on a finite rank deep kernel learning model. As described above, a finite rank deep kernel learning model may be used to forecast financial transactions, such as a user&#39;s financial transactions over time, concurrently with a prediction of the confidence of those forecasts. 
       FIG. 10A  depicts a table of historical user financial transaction data  1002 , which includes data related to multiple users (e.g., U 1 -U 3 ), a range of dates of transactions, and transaction amounts. The data in table  1002  may be processed by a finite rank deep kernel leaning model to produce output including forecasted values as well as an uncertainty measure for the forecasted values as depicted in table  1004 . The uncertainty measure may be used to form a confidence interval, as depicted in  FIG. 10B , which depicts forecasted data for one of the users in tables  1002  and  1004 . In this example, confidence interval  1010  is a 95% confidence interval, but in other examples, it may be any other confidence interval. 
       FIG. 10B  depicts a graphical forecast output showing actual data  1006  as well as the forecasted data  1008  with a confidence interval  1010 . As seen here again, despite having an irregular geometry, the finite rank deep kernel learning model is able to accurately model the actual data while providing simultaneous and time-dependent uncertainty quantification. 
     The prediction of financial transactions (e.g., cash flow) over time along with confidence intervals can improve the types of financial services offered to users. This is especially true where an organization wants to limit “false positive” alerts, such as an alert that the user&#39;s predicted cash flow would underrun their current bank account balance. Thus, the confidence interval can improve the ability to tune such alerts for user experience. 
     A second example application is forecasting computer resource utilization, including local and cloud-based resources. Computer resource utilization is particularly challenging to forecast over longer periods of time given it has cyclical and non-cyclical elements in addition to significant variability at any given time. This sort of complex geometry time series data is again an excellent candidate for analysis by a finite rank deep kernel learning model. In particular, in addition to creating better forecasts of resource utilization (and therefore needs) as compared to conventional modelling methods, finite rank deep kernel learning provides more accurate confidence intervals on the forecasts, which allows for strategic planning. For example, accurately forecasting computing resource utilization may create the opportunity to contract for cloud-based resources well in advance for better prices than spot prices for on-demand needs. Similarly, accurately forecasting computing resource utilization may create the opportunity to plan for resource expansion and allocate budget over a longer period of time. Further yet, the use of finite rank deep kernel learning may be a contributing factor in reducing resource utilization based on its significantly more efficient performance characteristics as compared to conventional modelling methods. 
     A third example application is detecting anomalous behavior to identify security risks e.g., from user log data or cloud based application logs. By nature, the more data captured regarding user behavior provides for more opportunities for detecting anomalies. But historically, significantly increasing the amount of captured user data (e.g., log data), meant likewise significantly increasing the amount of data that needed processing, the time to process it, the time to train models, etc. However, by using a computationally efficient method such as finite rank deep kernel learning, more data can be captured and processed without the conventional unsustainable increase in processing demand. Further, finite rank deep kernel learning creates more accurate confidence bounds, the detection accuracy (e.g., of anomalous behavior) is improved because the confidence of what is and is not anomalous is likewise improved. 
     A fourth example application is resource planning, for example, for online customer support. Some organizations may have significantly variable seasonal human resource needs. By way of example, a tax preparation organization may require significantly more human resources during tax preparation season as compared to the off-season. However, this sort of need is difficult to forecast given the complex nature of the underlying data. Finite rank deep kernel learning is well-suited for this task because it can learn with significant granularity local patterns in data sets. For example, as described above, the dot kernels provide a means for capturing localized data trends in a dataset while still creating a forecast that matches the characteristics of the dataset as a whole. 
     Many other example applications exist. Because the aforementioned application examples all likely involve very large data sets, the finite rank deep kernel learning method disclosed herein would significant improve processing system performance in terms of processing cycles, total cycle time, memory usage, and others. 
     Example Processing System 
       FIG. 11  depicts an example processing system  1100  for performing finite rank deep kernel learning. For example, processing system  1100  may be configured to perform one or more aspects of flow  700  described with respect to  FIG. 7  and method  900  described with respect to  FIG. 9 . 
     Processing system  1100  includes a CPU  1102  connected to a data bus  1130 . CPU  1102  is configured to process computer-executable instructions, e.g., stored in memory  1110  or storage  1120 , and to cause processing system  1100  to perform methods as described herein, for example with respect to  FIG. 9 . CPU  1102  is included to be representative of a single CPU, multiple CPUs, a single CPU having multiple processing cores, and other forms of processing architecture capable of executing computer-executable instructions. 
     Processing system  1100  further includes input/output device(s)  1104  and input/output interface(s)  1106 , which allow processing system  1100  to interface with input/output devices, such as, for example, keyboards, displays, mouse devices, pen input, and other devices that allow for interaction with processing system  1100 . 
     Processing system  1100  further includes network interface  1108 , which provides processing system  1100  with access to external networks, such as network  1114 . 
     Processing system  1100  further includes memory  1110 , which in this example includes a plurality of components. 
     For example, memory  1110  includes deep neural network component  1112 , which is configured to perform deep neural network functions as described above. 
     Memory  1110  further includes embedding component  1114 , which is configured to determine embeddings based on output from neural deep neural network component  1112 . For example, embedding component  1114  may identify orthogonal embeddings for use in creating a plurality of dot kernels. 
     Memory  1110  further includes dot kernel component  1116 , which is configured to determine dot kernels based on the embeddings determined by embedding component  1114 . 
     Memory  1110  further includes composite kernel component  1118 , which is configured to create composite (i.e., expressive) kernels from the dot kernels determined by dot kernel component  1116 . For example, composite kernel component  1118  may be configured to linearly combine a plurality of dot kernels to determine a composite kernel. 
     Note that while shown as a single memory  1110  in  FIG. 11  for simplicity, the various aspects stored in memory  1110  may be stored in different physical memories, but all accessible CPU  1102  via internal data connections, such as bus  1130 . 
     Processing system  1100  further includes storage  1120 , which in this example includes training data  1122 , live data  1124 , and model parameters  1126 . Training data  1122  may be, as described above, data used to train a finite rank deep kernel learning model. Live data  1124  may be data provided, for example, by an application, which is to be acted upon by the finite rank deep kernel leaning model. Model parameters  1126  may be parameters related to, for example, the deep neural network used to determine embeddings, as described above. 
     While not depicted in  FIG. 11 , other aspects may be included in storage  1110 . 
     As with memory  1110 , a single storage  1120  is depicted in  FIG. 11  for simplicity, but the various aspects stored in storage  1120  may be stored in different physical storages, but all accessible to CPU  1102  via internal data connections, such as bus  1130 , or external connection, such as network interface  1108 . 
     The preceding description is provided to enable any person skilled in the art to practice the various embodiments described herein. The examples discussed herein are not limiting of the scope, applicability, or embodiments set forth in the claims. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments. For example, changes may be made in the function and arrangement of elements discussed without departing from the scope of the disclosure. Various examples may omit, substitute, or add various procedures or components as appropriate. For instance, the methods described may be performed in an order different from that described, and various steps may be added, omitted, or combined. Also, features described with respect to some examples may be combined in some other examples. For example, an apparatus may be implemented or a method may be practiced using any number of the aspects set forth herein. In addition, the scope of the disclosure is intended to cover such an apparatus or method that is practiced using other structure, functionality, or structure and functionality in addition to, or other than, the various aspects of the disclosure set forth herein. It should be understood that any aspect of the disclosure disclosed herein may be embodied by one or more elements of a claim. 
     As used herein, the word “exemplary” means “serving as an example, instance, or illustration.” Any aspect described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects. 
     As used herein, a phrase referring to “at least one of” a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover a, b, c, a-b, a-c, b-c, and a-b-c, as well as any combination with multiples of the same element (e.g., a-a, a-a-a, a-a-b, a-a-c, a-b-b, a-c-c, b-b, b-b-b, b-b-c, c-c, and c-c-c or any other ordering of a, b, and c). 
     As used herein, the term “determining” encompasses a wide variety of actions. For example, “determining” may include calculating, computing, processing, deriving, investigating, looking up (e.g., looking up in a table, a database or another data structure), ascertaining and the like. Also, “determining” may include receiving (e.g., receiving information), accessing (e.g., accessing data in a memory) and the like. Also, “determining” may include resolving, selecting, choosing, establishing and the like. 
     The methods disclosed herein comprise one or more steps or actions for achieving the methods. The method steps and/or actions may be interchanged with one another without departing from the scope of the claims. In other words, unless a specific order of steps or actions is specified, the order and/or use of specific steps and/or actions may be modified without departing from the scope of the claims. Further, the various operations of methods described above may be performed by any suitable means capable of performing the corresponding functions. The means may include various hardware and/or software component(s) and/or module(s), including, but not limited to a circuit, an application specific integrated circuit (ASIC), or processor. Generally, where there are operations illustrated in figures, those operations may have corresponding counterpart means-plus-function components with similar numbering. 
     The various illustrative logical blocks, modules and circuits described in connection with the present disclosure may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device (PLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but in the alternative, the processor may be any commercially available processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. 
     A processing system may be implemented with a bus architecture. The bus may include any number of interconnecting buses and bridges depending on the specific application of the processing system and the overall design constraints. The bus may link together various circuits including a processor, machine-readable media, and input/output devices, among others. A user interface (e.g., keypad, display, mouse, joystick, etc.) may also be connected to the bus. The bus may also link various other circuits such as timing sources, peripherals, voltage regulators, power management circuits, and other circuit elements that are well known in the art, and therefore, will not be described any further. The processor may be implemented with one or more general-purpose and/or special-purpose processors. Examples include microprocessors, microcontrollers, DSP processors, and other circuitry that can execute software. Those skilled in the art will recognize how best to implement the described functionality for the processing system depending on the particular application and the overall design constraints imposed on the overall system. 
     If implemented in software, the functions may be stored or transmitted over as one or more instructions or code on a computer-readable medium. Software shall be construed broadly to mean instructions, data, or any combination thereof, whether referred to as software, firmware, middleware, microcode, hardware description language, or otherwise. Computer-readable media include both computer storage media and communication media, such as any medium that facilitates transfer of a computer program from one place to another. The processor may be responsible for managing the bus and general processing, including the execution of software modules stored on the computer-readable storage media. A computer-readable storage medium may be coupled to a processor such that the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. By way of example, the computer-readable media may include a transmission line, a carrier wave modulated by data, and/or a computer readable storage medium with instructions stored thereon separate from the wireless node, all of which may be accessed by the processor through the bus interface. Alternatively, or in addition, the computer-readable media, or any portion thereof, may be integrated into the processor, such as the case may be with cache and/or general register files. Examples of machine-readable storage media may include, by way of example, RAM (Random Access Memory), flash memory, ROM (Read Only Memory), PROM (Programmable Read-Only Memory), EPROM (Erasable Programmable Read-Only Memory), EEPROM (Electrically Erasable Programmable Read-Only Memory), registers, magnetic disks, optical disks, hard drives, or any other suitable storage medium, or any combination thereof. The machine-readable media may be embodied in a computer-program product. 
     A software module may comprise a single instruction, or many instructions, and may be distributed over several different code segments, among different programs, and across multiple storage media. The computer-readable media may comprise a number of software modules. The software modules include instructions that, when executed by an apparatus such as a processor, cause the processing system to perform various functions. The software modules may include a transmission module and a receiving module. Each software module may reside in a single storage device or be distributed across multiple storage devices. By way of example, a software module may be loaded into RAM from a hard drive when a triggering event occurs. During execution of the software module, the processor may load some of the instructions into cache to increase access speed. One or more cache lines may then be loaded into a general register file for execution by the processor. When referring to the functionality of a software module, it will be understood that such functionality is implemented by the processor when executing instructions from that software module. 
     The following claims are not intended to be limited to the embodiments shown herein, but are to be accorded the full scope consistent with the language of the claims. Within a claim, reference to an element in the singular is not intended to mean “one and only one” unless specifically so stated, but rather “one or more.” Unless specifically stated otherwise, the term “some” refers to one or more. No claim element is to be construed under the provisions of 35 U.S.C. § 112(f) unless the element is expressly recited using the phrase “means for” or, in the case of a method claim, the element is recited using the phrase “step for.” All structural and functional equivalents to the elements of the various aspects described throughout this disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims. Moreover, nothing disclosed herein is intended to be dedicated to the public regardless of whether such disclosure is explicitly recited in the claims.