Qt/C++特征分解eig奇异值分解SVD库

# Eigen Tensors Tensors are multidimensional arrays of elements. Elements are typically scalars, but more complex types such as strings are also supported. [TOC] ## Tensor Classes You can manipulate a tensor with one of the following classes. They all are in the namespace ```::Eigen.``` ### Class Tensor<data_type, rank> This is the class to use to create a tensor and allocate memory for it. The class is templatized with the tensor datatype, such as float or int, and the tensor rank. The rank is the number of dimensions, for example rank 2 is a matrix. Tensors of this class are resizable. For example, if you assign a tensor of a different size to a Tensor, that tensor is resized to match its new value. #### Constructor Tensor<data_type, rank>(size0, size1, ...) Constructor for a Tensor. The constructor must be passed ```rank``` integers indicating the sizes of the instance along each of the the ```rank``` dimensions. // Create a tensor of rank 3 of sizes 2, 3, 4. This tensor owns // memory to hold 24 floating point values (24 = 2 x 3 x 4). Tensor<float, 3> t_3d(2, 3, 4); // Resize t_3d by assigning a tensor of different sizes, but same rank. t_3d = Tensor<float, 3>(3, 4, 3); #### Constructor Tensor<data_type, rank>(size_array) Constructor where the sizes for the constructor are specified as an array of values instead of an explicitly list of parameters. The array type to use is ```Eigen::array<Eigen::Index>```. The array can be constructed automatically from an initializer list. // Create a tensor of strings of rank 2 with sizes 5, 7. Tensor<string, 2> t_2d({5, 7}); ### Class TensorFixedSize<data_type, Sizes<size0, size1, ...>> Class to use for tensors of fixed size, where the size is known at compile time. Fixed sized tensors can provide very fast computations because all their dimensions are known by the compiler. FixedSize tensors are not resizable. If the total number of elements in a fixed size tensor is small enough the tensor data is held onto the stack and does not cause heap allocation and free. // Create a 4 x 3 tensor of floats. TensorFixedSize<float, Sizes<4, 3>> t_4x3; ### Class TensorMap<Tensor<data_type, rank>> This is the class to use to create a tensor on top of memory allocated and owned by another part of your code. It allows to view any piece of allocated memory as a Tensor. Instances of this class do not own the memory where the data are stored. A TensorMap is not resizable because it does not own the memory where its data are stored. #### Constructor TensorMap<Tensor<data_type, rank>>(data, size0, size1, ...) Constructor for a Tensor. The constructor must be passed a pointer to the storage for the data, and "rank" size attributes. The storage has to be large enough to hold all the data. // Map a tensor of ints on top of stack-allocated storage. int storage[128]; // 2 x 4 x 2 x 8 = 128 TensorMap<Tensor<int, 4>> t_4d(storage, 2, 4, 2, 8); // The same storage can be viewed as a different tensor. // You can also pass the sizes as an array. TensorMap<Tensor<int, 2>> t_2d(storage, 16, 8); // You can also map fixed-size tensors. Here we get a 1d view of // the 2d fixed-size tensor. Tensor<float, Sizes<4, 5>> t_4x3; TensorMap<Tensor<float, 1>> t_12(t_4x3, 12); #### Class TensorRef See Assigning to a TensorRef below. ## Accessing Tensor Elements #### <data_type> tensor(index0, index1...) Return the element at position ```(index0, index1...)``` in tensor ```tensor```. You must pass as many parameters as the rank of ```tensor```. The expression can be used as an l-value to set the value of the element at the specified position. The value returned is of the datatype of the tensor. // Set the value of the element at position (0, 1, 0); Tensor<float, 3> t_3d(2, 3, 4); t_3d(0, 1, 0) = 12.0f; // Initialize all elements to random values. for (int i = 0; i < 2; ++i) { for (int j = 0; j < 3; ++j) { for (int k = 0; k < 4; ++k) { t_3d(i, j, k) = ...some random value...; } } } // Print elements of a tensor. for (int i = 0; i < 2; ++i) { LOG(INFO) << t_3d(i, 0, 0); } ## TensorLayout The tensor library supports 2 layouts: ```ColMajor``` (the default) and ```RowMajor```. Only the default column major layout is currently fully supported, and it is therefore not recommended to attempt to use the row major layout at the moment. The layout of a tensor is optionally specified as part of its type. If not specified explicitly column major is assumed. Tensor<float, 3, ColMajor> col_major; // equivalent to Tensor<float, 3> TensorMap<Tensor<float, 3, RowMajor> > row_major(data, ...); All the arguments to an expression must use the same layout. Attempting to mix different layouts will result in a compilation error. It is possible to change the layout of a tensor or an expression using the ```swap_layout()``` method. Note that this will also reverse the order of the dimensions. Tensor<float, 2, ColMajor> col_major(2, 4); Tensor<float, 2, RowMajor> row_major(2, 4); Tensor<float, 2> col_major_result = col_major; // ok, layouts match Tensor<float, 2> col_major_result = row_major; // will not compile // Simple layout swap col_major_result = row_major.swap_layout(); eigen_assert(col_major_result.dimension(0) == 4); eigen_assert(col_major_result.dimension(1) == 2); // Swap the layout and preserve the order of the dimensions array<int, 2> shuffle(1, 0); col_major_result = row_major.swap_layout().shuffle(shuffle); eigen_assert(col_major_result.dimension(0) == 2); eigen_assert(col_major_result.dimension(1) == 4); ## Tensor Operations The Eigen Tensor library provides a vast library of operations on Tensors: numerical operations such as addition and multiplication, geometry operations such as slicing and shuffling, etc. These operations are available as methods of the Tensor classes, and in some cases as operator overloads. For example the following code computes the elementwise addition of two tensors: Tensor<float, 3> t1(2, 3, 4); ...set some values in t1... Tensor<float, 3> t2(2, 3, 4); ...set some values in t2... // Set t3 to the element wise sum of t1 and t2 Tensor<float, 3> t3 = t1 + t2; While the code above looks easy enough, it is important to understand that the expression ```t1 + t2``` is not actually adding the values of the tensors. The expression instead constructs a "tensor operator" object of the class TensorCwiseBinaryOp<scalar_sum>, which has references to the tensors ```t1``` and ```t2```. This is a small C++ object that knows how to add ```t1``` and ```t2```. It is only when the value of the expression is assigned to the tensor ```t3``` that the addition is actually performed. Technically, this happens through the overloading of ```operator=()``` in the Tensor class. This mechanism for computing tensor expressions allows for lazy evaluation and optimizations which are what make the tensor library very fast. Of course, the tensor operators do nest, and the expression ```t1 + t2 * 0.3f``` is actually represented with the (approximate) tree of operators: TensorCwiseBinaryOp<scalar_sum>(t1, TensorCwiseUnaryOp<scalar_mul>(t2, 0.3f)) ### Tensor Operations and C++ "auto" Because Tensor operations create tensor operators, the C++ ```auto``` keyword does not have its intuitive meaning. Consider these 2 lines of code: Tensor<float, 3> t3 = t1 + t2; auto t4 = t1 + t2; In the first line we allocate the tensor ```t3``` and it will contain the result of the addition of ```t1``` and ```t2```. In the second line, ```t4``` is actually the tree of tensor operators that will compute the addition of ```t1``` and ```t2```. In fact, ```t4``` is *not* a tensor and you cannot get the values of its elements: Tensor<float, 3> t3 = t1 + t2