Understanding Data Representation, Approximation, and Operations in high Dimensional Spaces  

Euclidean Space:

Euclidean space refers to a specific type of linear space known as n-dimensional Euclidean space. It is characterized by its metric properties, where distances between points are defined using the Euclidean distance formula, and angles between vectors are defined using the dot product. In two-dimensional (2D) Euclidean space, a vector is represented by (x, y) coordinates, while in three-dimensional (3D) space, the coordinates expand to (x, y, z) to describe a point's position.

Vector Space:

A vector space is a more general term used to describe any linear space, regardless of its dimensionality. It encompasses both finite and infinite-dimensional spaces and includes Euclidean space as a special case when considering 2D or 3D space. In a vector space, vectors can be added and scaled, and they obey specific axioms that define the space's algebraic properties. In another word, it's a space where vectors can live abides to vector algebraic addition and multiplication properties. 

Here are some examples of vector spaces:

• Real Numbers (R): The set of all real numbers is a vector space. The operations of addition and scalar multiplication in this vector space are just ordinary addition and multiplication of real numbers.

• Euclidean Space (R^n): The n-dimensional Euclidean space, denoted R^n, is a vector space. This space consists of all ordered n-tuples of real numbers. The operations of vector addition and scalar multiplication are performed component-wise.

• Polynomial Space (Pn): The space of all polynomials of degree less than or equal to n, denoted Pn, is a vector space. The operations of vector addition and scalar multiplication are performed term-by-term.

For example, P_2 is the set of all polynomials of degree less than or equal to 2. It includes functions such as:

• f(x) = x^2

• g(x) = 2x + 1

• h(x) = 3

In this space, we can perform operations like addition, subtraction and scalar multiplication, similar to other vector spaces. Here are a few examples:

• (f + g)(x) = f(x) + g(x) = x^2 + 2x + 1

• (f - h)(x) = f(x) - h(x) = x^2 - 3

• (2f)(x) = 2 * f(x) = 2x^2

Moreover, the zero vector in this space is the zero polynomial, the polynomial p(x) = 0 for all x.

• Function Space (C[a, b]): The space of all continuous real-valued functions defined on the interval [a, b] is a vector space. Vector addition and scalar multiplication are defined as the pointwise addition and multiplication of functions.

For example, consider the set of all real-valued functions of a real variable, denoted by F(R, R). This means that each function in this set takes a real number as input and produces a real number as output.

The function f(x) = x^2 and g(x) = sin(x) are both elements of this function space. We can perform operations such as addition and scalar multiplication in this function space. For instance:

• (f + g)(x) = f(x) + g(x) = x^2 + sin(x)

• (cf)(x) = c * f(x) = c * x^2 for any real number c.

Fourier Series

The Fourier basis is associated with a function space, specifically the space of functions that are square-integrable over a certain interval (often taken to be [-π, π] or [0, 2π]). The concept of a Fourier series tells us that any function in our function space can be written as a (possibly infinite) linear combination of sine and cosine functions, which serve as the Fourier basis functions. 

Another common example of a function space is the space of square-integrable functions, denoted by L^2, which is used frequently in quantum mechanics 

• • The function f(x) = e^(-x^2) is square-integrable. To see this, we need to check that the integral of |f(x)|^2 from -infinity to infinity is finite:
    Integral from -∞ to ∞ of |e^(-x^2)|^2 dx = Integral from -∞ to ∞ of e^(-2x^2) dx = sqrt(pi), which is finite.

  • The function g(x) = sin(x) is also square-integrable. To see this, we need to check that the integral of |g(x)|^2 from -infinity to infinity is finite:
    Integral from -∞ to ∞ of |sin(x)|^2 dx = Integral from -∞ to ∞ of (1/2 - 1/2 cos(2x)) dx = ∞, which means we need to take a finite interval.
    If we take the interval from 0 to pi, then:
    Integral from 0 to π of (1/2 - 1/2 cos(2x)) dx = π/2, which is finite.

• Matrix Space (Mn, m): The space of all n by m matrices, denoted Mn, m, is a vector space. Vector addition and scalar multiplication are performed element-wise.

A matrix space is the set of all matrices of a particular dimension. Here's an example:

Let's consider the space of 2x2 real matrices. This space is denoted as M_2,2(R), and it is a vector space. A typical element of this space looks like:

[a, b]

[c, d]

where a, b, c, d are real numbers.

The vector addition and scalar multiplication in this space are defined as follows:

• Vector addition:
  If we have two matrices A and B in this space:
  A = [a1, b1] [c1, d1]
  B = [a2, b2] [c2, d2]
  Then, A + B is computed element-wise:
  A + B = [a1 + a2, b1 + b2] [c1 + c2, d1 + d2]

• Scalar multiplication:
  If we have a matrix A and a scalar λ (a real number), the scalar multiplication is also performed element-wise:
  λA = λ[a1, b1] = [λa1, λb1] [λc1, λd1]

So, the space M_2,2(R) is a vector space because it includes the zero vector (the 2x2 zero matrix), it is closed under vector addition and scalar multiplication, and these operations satisfy the required properties for a vector space.

These examples show the variety of sets that can form vector spaces, from the familiar real numbers and n-dimensional spaces to more abstract spaces of functions, polynomials, and matrices. Each of these vector spaces can be studied with the tools of linear algebra.

Subspace of a Real Linear Vector Space: 

A subspace is a subset of a vector space that forms a vector space itself. It contains all the properties of a vector space, such as closure under vector addition and scalar multiplication. Subspaces are essential in various mathematical applications, including linear transformations and eigenspaces, which help understand data variance and patterns. Below is a depictions for projecting 3D points into respective 2D (A plane with the 3D space) and 1D subspaces (A line within the 3D space). 

Note: We want to emphasize that a subspace is represented by a hyperplane (plane in 2D and a line in 1D) that passes through the origin. This is crucial for linear transformation properties, especially when working with different coordinate systems or vector spaces involving translation and rotation. 

Hilbert Space: 

A Hilbert space is a particular type of vector space with additional mathematical structure. It is an infinite-dimensional space equipped with an inner product that allows for the definition of lengths and angles. Hilbert spaces are foundational in functional analysis, quantum mechanics, and signal processing, and they find applications in representing infinite-dimensional data, such as continuous signals and functions.

Normed Space: 

A vector space that is furnished with a norm is referred to as a normed space. A norm is a special function that, in a way coherent with the vector space's structure, attributes a strictly positive length or size to each vector in the space - the sole exception being the zero vector, to which it assigns a length of zero. An effective method to calculate the norm of a vector involves taking the square root of the inner product of the vector with itself, denoted as sqrt(<v, v>), where v is the vector. The introduction of a norm in a vector space endows it with the valuable properties of size and distance, which are instrumental in many mathematical and real-world applications, such as in defining metrics for error measurement, or distances in machine learning algorithms. .

Note: It's important to understand that any vector space can have a norm defined on it, but not all vector spaces do have a norm defined. Whether a vector space is a normed space depends on whether a norm has been explicitly defined for that space. 

For example, consider the vector space of all polynomials with real coefficients, denoted as P(R). We can add and scale polynomials, so this set of all polynomials forms a vector space. But if we haven't defined a norm (a measure of "length") for these polynomials, then this vector space is not a normed space. 

Metric Space: 

A metric space is another crucial concept in space representation. It is a set with a distance function (metric) defined on it, enabling the measurement of distances between elements. Metric spaces are fundamental in topology, analysis, and data science, as they provide a way to quantify and compare relationships between data points, aiding in clustering and similarity-based analysis.

Note: A normed space and a metric space are both mathematical constructs that, in essence, allow for a concept of "distance" between points. However, they differ in their structure and the level of strictness in their definitions. Here's a closer look:

Normed Space: A normed space is a vector space on which a norm is defined. A norm is a function that assigns a non-negative length or size to each vector in the vector space - with the zero vector being the only vector assigned a length of zero - in a way that aligns with the structure of the vector space. This norm function must satisfy certain properties, including definiteness, scalability or absolute homogeneity, and the triangle inequality.

Metric Space: A metric space is a set equipped with a function called a metric. This metric, denoted as d(x, y) where x and y are elements of the set, defines the "distance" between any two points in the set. Like a norm, a metric must satisfy certain properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

The key differences between them are:

• Every normed space is a metric space if we define the distance between two vectors as the norm of their difference: d(x, y) = ||x - y||. However, not every metric space is a normed space, because a metric doesn't have to comply with the properties of scalability and compatibility with vector addition that a norm does.

• A normed space is a vector space, meaning it has vector addition and scalar multiplication operations, while a metric space only has a concept of distance, and doesn't necessarily support vector space operations.

• The triangle inequality in a metric space involves distances between points, while the triangle inequality in a normed space involves vector addition and scalar multiplication.

To summarize, while both normed spaces and metric spaces give us a way to measure distance, a normed space is a more specific structure that also allows us to perform vector operations and measure vector lengths

Complete Space:

A metric space (or normed space) is said to be complete if every Cauchy sequence in the space converges to a point in the space. A Cauchy sequence is a sequence where the distance between its terms becomes arbitrarily small as the sequence progresses.

Banach Space:

Putting it all together, a Banach space is a normed vector space that is complete. This means not only can we talk about the size or length of vectors and the distance between them, but every sequence of vectors that "should" converge (i.e., is Cauchy) does converge to a vector in the space. In a Banach space, the idea of limits and convergence is well-defined for all sequences that intuitively should have limits. This property ensures that when we're discussing the action of operators (systems) on signals and the convergence behavior of sequences of signals, everything behaves as expected. 

In essence, the completeness property of Banach spaces provides a robust framework ensuring that sequences converge, operators behave predictably, and various mathematical constructs have the properties we expect. This assurance is not guaranteed in more general metric or normed spaces. Without this completeness, many of the powerful results and theorems in functional analysis wouldn't hold, or their proofs would be far more intricate. 

Kernel Space:

Kernel space is an important addition to the discussion of space representation, particularly in machine learning and data analysis. It extends the concept of Euclidean space to represent non-linear relationships between data points. By utilizing a kernel function, it computes the inner product between data points in a higher-dimensional feature space. This technique effectively transforms data points, making them linearly separable in higher dimensions, even when they are not separable in the original Euclidean space. Kernel methods, like the kernel trick in Support Vector Machines (SVMs), find applications in diverse areas, such as classification, regression, and dimensionality reduction tasks.

Mathematical Spaces Summary

• Vector Space: The most basic type of space. It's a set of vectors that is closed under vector addition and scalar multiplication.

• Inner Product Space: A vector space with an additional structure called the "inner product," which allows you to measure angles between vectors.

• Hilbert Space: An inner product space that is also complete in the sense that any Cauchy sequence of vectors has a limit that is also in the space.

• Normed Space: A vector space equipped with a norm, which measures the length or size of vectors.

• Banach Space: A normed space that is complete, similar to Hilbert spaces but without requiring an inner product.

• Metric Space: A more general space that doesn't have to be a vector space. It is equipped with a metric to measure distances between points.

Linear Space Construction and Orthogonal Basis:

To construct a linear space, one can start with a set of vectors and extend it using linear combinations of these vectors. An orthogonal basis is a special type of basis that consists of linearly independent vectors that are mutually orthogonal (perpendicular). Having an orthogonal basis simplifies vector operations and allows for a more efficient representation of the space.

Nonlinear Space and Manifold:

In contrast to linear spaces, nonlinear spaces lack the properties of vector addition and scalar multiplication. A manifold is a prime example of a nonlinear space, resembling Euclidean space locally. Manifolds find applications in geometry, physics, and data representation. They are crucial because projecting or representing objects or information from one space to a subspace might not be optimally captured using a linear entity (like a line or plane in lower-dimensional spaces). Instead, a more suitable representation could be a smooth-surfaced spherical object, like a sphere, which preserves most of the visual information. For instance, projecting a 3D world view into a panoramic representation or mapping the same information onto a 2D nonlinear manifold, such as a sphere, can be more effective in preserving essential details. 

Important Concepts of High dimensional Information Construction and Properties:

Vector Norm:

A vector norm is a function that assigns a non-negative length or size to a vector. It satisfies certain properties, such as non-negativity, homogeneity, and the triangle inequality. Common examples of vector norms include the Euclidean norm (also known as the 2-norm) and the Manhattan norm (also known as the 1-norm).

Orthonormal:

A set of vectors is orthonormal if all the vectors are mutually orthogonal and have a unit norm (length equal to 1). Orthonormal vectors are often used in linear algebra and signal processing due to their mathematical simplicity and usefulness in computations.

Operations in Vector Space:

Just like we can addition, multiplication can transform data in classic algebra. For vector, there are also mathmatic operations that we can apply to to transform these data.

Inner Product Operator:

The inner product operator is a mathematical operation that takes two vectors from a vector space and produces a scalar value. It measures the "closeness" or alignment between the two vectors and is defined using the dot product in Euclidean space or the general inner product in a more abstract vector space.

Multiplication Operator:

The multiplication operator is a linear transformation that multiplies a vector by a fixed matrix or scalar. It is a fundamental operation in linear algebra, and its applications range from transformations in graphics to solving systems of linear equations.

• Matrix-Vector Multiplication Product is the result of the multiplication. in general, it's result of the multiplication Ax = B, where A is an mxn matrix, x is an N-dimensional vector, and B is an m-dimensional vector. The number of columns in A (n) must be equal to the dimension of x (N) for the multiplication to be valid. 

Note. The equation Ax = B represents a system of linear equations that can be solved to find the values of vector x that satisfy the equation for the given vector B. Matrix A is a linear transformation matrix that maps data X from the input vector space to the output vector space, which is spanned by the column vectors of A. Each column vector of A represents a basis vector in the output vector space.

Addition Operator:

The addition operator, simply put, is the operation of adding two vectors element-wise to create a new vector that represents their sum. It is a fundamental operation in linear spaces and plays a central role in vector calculations.

Adjoint Operator:

In the context of linear transformations, the adjoint operator, also known as the Hermitian adjoint or conjugate transpose, is the generalization of the transpose for complex vector spaces. It is used to represent the transpose of a matrix or the complex conjugate transpose.

Projection Operator:

A projection operator is a linear transformation that projects a vector onto a subspace along a specified direction. It is commonly used in signal processing, image compression, and data reduction techniques.

Subspace Methods for Approximating Higher-dimensional Objects into Lower-dimensional Presentations:

Least Square Approximation:

Least square approximation is a method used to find the best-fitting linear representation of data when there is no exact solution. It minimizes the sum of squared errors between the data points and the linear model.

Orthogonal Projection:

Orthogonal projection is a technique used to project a vector onto a subspace along a direction that is perpendicular (orthogonal) to that subspace. It is particularly useful when dealing with orthogonal bases.

Pseudoinverse:

The pseudoinverse, also known as the Moore-Penrose inverse, is a generalization of the matrix inverse to non-square and singular matrices. It is used in cases where the matrix is not invertible and helps in solving linear systems with more equations than unknowns.

SVD (Singular Value Decomposition):

SVD is a powerful matrix factorization technique that represents a matrix as a product of three matrices: U, Σ, and V^T (transpose of V). It is widely used in data compression, image processing, and collaborative filtering algorithms.

Practical Example 1 - Hand Gesture Recognition:

Let's consider a practical example to illustrate the concepts discussed above. Imagine you are working on a computer vision project where you need to recognize hand gestures to control a virtual game. To achieve this, you have a camera that captures images of the hand in real-time.

1. Space Representation: Each image can be thought of as a point in a high-dimensional space, where each pixel's intensity corresponds to a specific coordinate in color space (i.e R, G B, coordinate system). Since the images are high-dimensional, directly processing them can be computationally expensive and challenging.

2. Dimensionality Reduction: To reduce the dimensionality and capture the essential features of the hand gestures, you can use the Singular Value Decomposition (SVD) method. SVD will decompose the high-dimensional image data into three matrices: U, Σ, and V^T. The matrix U represents the orthonormal basis of the image space, while Σ contains the singular values, and V^T captures the projection of images onto the new basis.

3. Approximation: By retaining only the most significant singular values and corresponding basis vectors, you can approximate each image using a lower-dimensional representation. This approximation will enable you to represent complex hand gestures with fewer parameters, making the recognition task more efficient.

4. Subspace Classification: After reducing the dimensionality, you can define a subspace for each specific hand gesture (e.g., open palm, fist, thumbs-up) based on the low-dimensional representations. During real-time recognition, you can project new incoming images onto each subspace using the projection operator.

5. Classification: To classify the hand gesture, you can use the inner product operator to calculate the similarity between the projected image and the subspace representation of each gesture. The gesture with the highest similarity score will be identified as the recognized hand gesture.

Practical Example 2 - Flower Classification:

• Data Representation: Each image of a flower can be thought of as a point in a high-dimensional space, where each pixel's intensity can be represented by a color space, such as RGB (Red, Green, Blue) or in another word, the pixel intensity value represents the features of the images. The collection of these images forms a high-dimensional dataset.

• Dimensionality Reduction: To reduce the high dimensionality and extract essential features, you apply dimensionality reduction techniques like Singular Value Decomposition (SVD). SVD decomposes the dataset into a lower-dimensional representation, capturing the most significant variations in the images.

• Metric Space: With the reduced dimensional representation, you transform the dataset into a metric space. You employ a distance metric, such as the Euclidean distance, to measure the similarity or dissimilarity between images. This step enables the quantification of how closely related different flower species are based on their image features.

• Clustering: Using the distance metric in the metric space, you perform clustering algorithms to group similar flowers together. For instance, you might use k-means clustering to identify distinct clusters of flower species.

• Classification: Finally, you build a machine learning model using the clustered data. The model can identify new flower images and classify them into the corresponding clusters or species based on their similarity to the clustered data.

• Metrics Evaluation: You use various performance metrics, such as accuracy and precision, to evaluate the classification model's effectiveness in recognizing different flower species.

The visualization showcases a comparison between the original Iris dataset and its reduced version with K-means clustering. The left subplot represents the original dataset in a 3D space, where each data point corresponds to an Iris flower sample. The three axes represent the sepal length, sepal width, and petal length in centimeters, respectively. Different colors highlight the three distinct species of Iris flowers: setosa, versicolor, and virginica.

In contrast, the right subplot displays the reduced dataset after applying TruncatedSVD, representing the data in a 2D space. The two principal components, Component 1 and Component 2, capture the most significant patterns in the original dataset. K-means clustering has further segmented the data into three distinct clusters, indicated by varying colors.

The comparison allows us to observe the distribution of the Iris flower samples in the reduced 2D space compared to their original 3D representation. Although reduced to just two dimensions, the reduced dataset successfully retains essential patterns and species groupings, showcasing the power of dimensionality reduction techniques.