Hilbert Spaces and Inner Products
To understand how geometric structure aids machine learning, you need to grasp the concepts of inner product and Hilbert space. An inner product is a function that takes two elements (usually vectors or functions) from a vector space and returns a real number, capturing a notion of angle and length. For vectors x and y in Rn, the standard inner product is the dot product:
<x,y>=sum(xi∗yi).
A Hilbert space is a vector space equipped with an inner product where every Cauchy sequence converges in the space (that is, it is complete with respect to the norm induced by the inner product). This completeness property ensures that limits of "well-behaved" sequences of functions or vectors stay within the space, which is crucial for stability in learning algorithms.
A key example is the space L2(a,b), which consists of all square-integrable functions on the interval [a,b]. The inner product here is defined as <f,g>=∫abf(x)g(x)dx. In machine learning, such spaces allow you to treat hypotheses (such as functions learned by regression) as points in a geometric space, making concepts like distance, orthogonality, and projection meaningful and actionable.
A fundamental property of inner product spaces is the parallelogram law, which states that for any two elements x and y in the space, the following holds:
∣∣x+y∣∣2+∣∣x−y∣∣2=2∣∣x∣∣2+2∣∣y∣∣2This equation is not just a curiosity—it actually characterizes inner product spaces among normed spaces. If a norm satisfies the parallelogram law, then the norm must arise from some inner product.
Proof sketch:
Start by expanding the norms using the definition induced by the inner product:
- ∣∣x+y∣∣2=<x+y,x+y>=<x,x>+2<x,y>+<y,y>;
- ∣∣x−y∣∣2=<x−y,x−y>=<x,x>−2<x,y>+<y,y>. Adding these gives:
- ∣∣x+y∣∣2+∣∣x−y∣∣2=2<x,x>+2<y,y>=2∣∣x∣∣2+2∣∣y∣∣2.
The parallelogram law is essential in analysis and learning theory because it ensures that the geometry of the space behaves nicely, enabling the use of projections and decompositions that are central to many algorithms.
Orthogonality in a Hilbert space means that two elements have zero inner product: they are "perpendicular" in a geometric sense. This is more than just a formal property — it allows you to decompose hypotheses into independent components, much like breaking a vector into directions. Projections use orthogonality to find the "closest" point in a subspace to a given element, which is at the heart of least-squares regression and many learning algorithms. Understanding these concepts helps you see how functional-analytic structures enable efficient and interpretable solutions in machine learning.
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To understand how geometric structure aids machine learning, you need to grasp the concepts of inner product and Hilbert space. An inner product is a function that takes two elements (usually vectors or functions) from a vector space and returns a real number, capturing a notion of angle and length. For vectors x and y in Rn, the standard inner product is the dot product:
<x,y>=sum(xi∗yi).
A Hilbert space is a vector space equipped with an inner product where every Cauchy sequence converges in the space (that is, it is complete with respect to the norm induced by the inner product). This completeness property ensures that limits of "well-behaved" sequences of functions or vectors stay within the space, which is crucial for stability in learning algorithms.
A key example is the space L2(a,b), which consists of all square-integrable functions on the interval [a,b]. The inner product here is defined as <f,g>=∫abf(x)g(x)dx. In machine learning, such spaces allow you to treat hypotheses (such as functions learned by regression) as points in a geometric space, making concepts like distance, orthogonality, and projection meaningful and actionable.
A fundamental property of inner product spaces is the parallelogram law, which states that for any two elements x and y in the space, the following holds:
∣∣x+y∣∣2+∣∣x−y∣∣2=2∣∣x∣∣2+2∣∣y∣∣2This equation is not just a curiosity—it actually characterizes inner product spaces among normed spaces. If a norm satisfies the parallelogram law, then the norm must arise from some inner product.
Proof sketch:
Start by expanding the norms using the definition induced by the inner product:
- ∣∣x+y∣∣2=<x+y,x+y>=<x,x>+2<x,y>+<y,y>;
- ∣∣x−y∣∣2=<x−y,x−y>=<x,x>−2<x,y>+<y,y>. Adding these gives:
- ∣∣x+y∣∣2+∣∣x−y∣∣2=2<x,x>+2<y,y>=2∣∣x∣∣2+2∣∣y∣∣2.
The parallelogram law is essential in analysis and learning theory because it ensures that the geometry of the space behaves nicely, enabling the use of projections and decompositions that are central to many algorithms.
Orthogonality in a Hilbert space means that two elements have zero inner product: they are "perpendicular" in a geometric sense. This is more than just a formal property — it allows you to decompose hypotheses into independent components, much like breaking a vector into directions. Projections use orthogonality to find the "closest" point in a subspace to a given element, which is at the heart of least-squares regression and many learning algorithms. Understanding these concepts helps you see how functional-analytic structures enable efficient and interpretable solutions in machine learning.
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