Enhancing Tensor Calculations In R Proposal For A Thin Abstraction Layer
Introduction
I am writing to propose a thin tensor abstraction layer for your R package, aimed at enhancing the conciseness and safety of tensor calculation expressions. During my utilization of your package for tensor calculations, I developed this abstraction layer, which I believe could be a valuable addition. This layer introduces the concept of lower and upper indices, enabling differentiation between tensor contraction/Einstein summation and simple value selection with equal index values. While the automatic lowering and raising of indices with custom metric tensors is a potential future enhancement, it is not yet implemented. I believe this layer would seamlessly integrate into your package, and I am excited to share the core concepts with you. If you find them promising, I am prepared to submit pull requests. Let's delve into the details of this abstraction layer and explore its potential benefits for tensor calculations in R.
a <- array(1:(2^2*3), dim = c(2,2,3))
Creating a Tensor
a_{ij}^{\ \ k}
The core object within this abstraction is the tensor object, creatable using the %_% operator from an array. The index order is crucial, as it dictates the matching of indices to array dimensions. Employing a dedicated tensor class offers several advantages. It allows for the overloading of standard mathematical operators, providing greater flexibility and expressiveness in tensor calculations. Furthermore, it enables precise control over allowed operations, ensuring the integrity and validity of tensor manipulations. This is crucial for preventing errors and maintaining the consistency of results. The use of a dedicated tensor class also opens the door for future extensions and optimizations, such as custom printing methods and specialized storage strategies. This makes the tensor class a foundational element for robust and efficient tensor calculations within the R environment. Let's explore how we create these tensor objects and define their index structure.
# two lower indices, one upper index
# the hat "^" identifies an upper index. A "_" could be used to explicitly set the index to "low", as of now, no prefix also sets a lowered index.
a %_% c("i", "j", "^k")
#> Tensor (2,2,3)_i_j^k #12
Implicit Contraction
b_k=a_{i\;k}^{\;i}
Tensor contraction, also known as Einstein summation, is a fundamental operation in tensor algebra. It involves summing over repeated indices, one covariant (lower) and one contravariant (upper), in a tensor expression. This operation reduces the rank of the tensor, effectively projecting it onto a lower-dimensional space. The implicit contraction mechanism implemented in this abstraction layer significantly simplifies tensor calculations by automating the summation process. By simply specifying the indices with appropriate upper and lower designations, the contraction is performed automatically, eliminating the need for manual summation loops or complex indexing operations. This not only reduces the code's verbosity but also enhances its readability, making it easier to understand and verify the correctness of tensor expressions. The system efficiently identifies and contracts over paired upper and lower indices, allowing users to focus on the higher-level mathematical operations rather than the intricate details of index manipulation. Let's examine how implicit contraction is performed using our tensor abstraction layer.
a %_% c("i", "^i", "k")
#> Tensor (3)_k #3
#> [1] 5 13 21
Automatic contraction or Einstein summation is exclusively performed over paired lower-upper indices. The presence of identical indices in the same position results in a generalization of “taking the diagonal”, extracting elements where the specified indices have the same value. This behavior aligns with the mathematical definition of tensor contraction, where summation occurs over indices that appear both as subscripts (lower indices) and superscripts (upper indices). The restriction of contraction to paired lower-upper indices ensures that the operation adheres to the rules of tensor algebra, preventing unintended or mathematically incorrect summations. This careful handling of indices is crucial for maintaining the integrity of tensor calculations and producing meaningful results. Let's see how this differs from simply taking the diagonal.
c_{ik}=a_{iik}
a %_% c("i", "i", "k")
#> Tensor (2,3)_i_k #6
#> [,1] [,2] [,3]
#> [1,] 1 5 9
#> [2,] 4 8 12
Multiplication or Addition
Multiplication and addition operations are seamlessly integrated into this tensor abstraction, leveraging the standard operators for intuitive use. Einstein summation is automatically applied for upper-lower paired indices during multiplication, further streamlining tensor calculations. However, equal-positioned, equal-named indices are treated differently, resulting in the “taking the diagonal part” operation rather than summation. This distinction is crucial for maintaining mathematical correctness and providing flexibility in tensor manipulations. The system intelligently interprets the index structure and applies the appropriate operation, whether it's contraction, element-wise multiplication, or diagonal extraction. This level of automation significantly simplifies the process of working with tensors, allowing users to focus on the underlying mathematics rather than the intricacies of index manipulation. This clear distinction between Einstein summation and diagonal extraction enhances the expressiveness and safety of tensor operations within the abstraction layer. Let's explore multiplication and addition in more detail.
Multiplication
d_{ijklmn}=a_{ijk}a_{lmn}
Tensor multiplication, a cornerstone of tensor algebra, involves combining two tensors to produce a new tensor with a rank equal to the sum of the ranks of the original tensors. This operation can be performed in various ways, including the outer product and the tensor product. The abstraction layer elegantly handles tensor multiplication, automating Einstein summation for paired upper-lower indices. This automatic summation simplifies complex expressions, allowing users to focus on the higher-level mathematical operations. When multiplying tensors, the resulting tensor's dimensions are determined by the non-contracted indices of the input tensors. This ensures that the resulting tensor accurately reflects the mathematical outcome of the multiplication. The system intelligently manages the index structure, performing the necessary summations and arranging the resulting tensor elements in the correct order. This automation not only reduces the code's complexity but also minimizes the risk of errors in manual index manipulation. Let's examine how tensor multiplication is expressed within the abstraction layer.
a %_% c("i", "j", "k") * a %_% c("l", "m", "n")
#> Tensor (2,2,3,2,2,3)_i_j_k_l_m_n #144
e=a_{ijk}a^{ijk}
a %_% c("i", "j", "k") * a %_% c("^i", "^j", "^k")
#> Scalar
#> [1] 650
Addition
Addition of tensors is performed exclusively on tensors sharing the identical index structure, ensuring mathematical consistency and preventing erroneous operations. This restriction is fundamental to tensor algebra, as adding tensors with differing index structures would result in a mathematically undefined operation. The abstraction layer enforces this rule, ensuring that addition is only performed on compatible tensors. This strict type checking prevents common errors and maintains the integrity of tensor calculations. The requirement for identical index structures extends beyond the names of the indices; it also includes their positions (upper or lower) and the dimensions associated with each index. This comprehensive check guarantees that the tensors are mathematically compatible for addition. The system performs element-wise addition, resulting in a new tensor with the same index structure as the original tensors. This streamlined approach simplifies tensor addition, making it intuitive and less prone to errors. Let's see how addition works within the abstraction layer.
f_{ijk} = a_{ijk} + a_{ijk}
a %_% c("i", "j", "k") + a %_% c("j", "i", "k")
#> Tensor (2,2,3)_i_j_k #12
Adding tensors of different index structures is explicitly disallowed, reinforcing the mathematical rigor of the abstraction layer. This prohibition is a crucial safeguard against common errors in tensor algebra, where adding incompatible tensors can lead to meaningless results. The system's strict type checking ensures that only tensors with matching index structures are permitted to be added, preventing accidental misuse and maintaining the integrity of calculations. This design choice aligns with the fundamental principles of tensor algebra and promotes robust, reliable tensor operations. The error message generated when attempting to add incompatible tensors provides clear feedback to the user, guiding them towards the correct usage and preventing further complications. This focus on error prevention and informative feedback enhances the user experience and fosters a deeper understanding of tensor algebra concepts.
# a %_% c("i", "j", "k") + a %_% c("i", "^i", "k")
Comparison of Tensors
Tensors can be compared effectively by matching their index names, allowing for a logical assessment of their structural similarity. This comparison method focuses on the arrangement and labeling of indices, providing insights into the tensors' underlying mathematical properties. Tensors with identical index names in the same positions are considered structurally equivalent, indicating that they represent the same type of mathematical object. This type of comparison is crucial in various tensor operations, such as checking compatibility for addition or determining if two tensors represent the same physical quantity in different coordinate systems. The abstraction layer facilitates this comparison by providing a straightforward mechanism for matching index names, simplifying the process of analyzing tensor relationships. This feature enhances the usability of the package and promotes a deeper understanding of tensor algebra concepts. Let's explore how tensor comparison works within the abstraction layer.
a_{ijk} = a_{ijk}
a %_% c("i", "j", "k") == a %_% c("i", "j", "k")
#> [1] TRUE
while
a %_% c("i", "j", "k") == a %_% c("j", "i", "k")
#> [1] FALSE
a_{ijk} \neq a_{jik}
Order of Dimensions
Once a tensor object is created, an association between an index name and a specific dimension is established. Tensor object operations meticulously maintain this association, ensuring that the index-dimension mapping remains consistent throughout calculations. However, the internal dimension ordering, while crucial for the underlying implementation, is treated as an implementation detail and can change depending on the operations performed. This flexibility in internal ordering allows for optimizations and efficient execution of tensor operations. To retrieve a well-defined array, which is an object with a specific dimension ordering, one simply uses the as.array() function, including the desired index ordering as an argument. This provides a mechanism for extracting the tensor data in a predictable and controlled manner, regardless of the internal dimension arrangement. This separation of internal representation from external presentation enhances the usability of the abstraction layer and allows users to focus on the mathematical meaning of the tensors rather than the implementation details. Let's see how this works in practice.
# the internal dimension ordering is an implementation detail and
# can change
b <- a %_% c("i", "j", "k") * a %_% c("j", "i", "^k")
# but we can impose an order when extracting the array:
as.array(b, c("i", "j"))
#> [,1] [,2]
#> [1,] 107 158
#> [2,] 158 224
