Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Euclidean Space: Inner Products, Projections and the Margin

6 minute read

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TL;DR: Everything in Euclidean geometry descends from the inner product. The norm is \(\lVert x\rVert=\sqrt{\langle x,x\rangle}\), the angle is defined by \(\cos\theta = \langle x,y\rangle/(\lVert x\rVert\lVert y\rVert)\), and orthogonality is \(\langle x,y\rangle = 0\). Projecting \(x\) onto a direction \(u\) means choosing the coefficient that minimises the residual, which gives \(\langle x,u\rangle u/\lVert u\rVert^2\) and leaves a residual orthogonal to \(u\). Apply that to a hyperplane \(\mathbf{w}^\top x + b = 0\) and the distance from any point is \(\lvert \mathbf{w}^\top x_0 + b\rvert / \lVert \mathbf{w}\rVert\), which is the quantity an SVM maximises.

The one structure everything else comes from

A vector space on its own has no lengths and no angles; it only knows about addition and scaling. Add an inner product \(\langle \cdot,\cdot\rangle\), bilinear, symmetric, and positive definite (\(\langle x,x\rangle > 0\) for \(x \neq 0\)), and geometry appears. In \(\mathbb{R}^n\) the standard choice is \(\langle x,y\rangle = x^\top y\), and the norm it induces is

\[ \lVert x \rVert = \sqrt{\langle x, x\rangle} = \sqrt{\textstyle\sum_i x_i^2}. \]

The Cauchy–Schwarz inequality \(\lvert\langle x,y\rangle\rvert \le \lVert x\rVert\,\lVert y\rVert\) guarantees that the ratio \(\langle x,y\rangle/(\lVert x\rVert \lVert y\rVert)\) always lies in \([-1,1]\), so it is legitimate to define the angle \(\theta\) between two vectors by

\[ \cos\theta = \frac{\langle x, y\rangle}{\lVert x\rVert\,\lVert y\rVert}. \]

That definition is cosine similarity. It is not an analogy to angles; it is the angle. Orthogonality, \(\langle x,y\rangle = 0\), is the case \(\theta = 90^\circ\).

Projection, derived rather than quoted

Given \(x\) and a nonzero direction \(u\), which multiple of \(u\) is closest to \(x\)? Minimise the squared residual over the scalar \(\alpha\):

\[ \frac{d}{d\alpha}\,\lVert x - \alpha u\rVert^2 = \frac{d}{d\alpha}\left(\lVert x\rVert^2 - 2\alpha\langle x,u\rangle + \alpha^2\lVert u\rVert^2\right) = -2\langle x,u\rangle + 2\alpha\lVert u\rVert^2 = 0 . \]

So \(\alpha^\star = \langle x,u\rangle/\lVert u\rVert^2\) and the projection is \(\mathrm{proj}_u(x) = \frac{\langle x,u\rangle}{\lVert u\rVert^2}\,u\). The residual is automatically orthogonal to \(u\), since \(\langle x - \alpha^\star u,\ u\rangle = \langle x,u\rangle - \alpha^\star\lVert u\rVert^2 = 0\). Least squares is this same argument in higher dimensions: the fitted values are the projection of \(y\) onto the column space of the design matrix, and the residual vector is orthogonal to every regressor.

Projection of a vector onto a line, with orthogonal residualA two-dimensional plot. The vector x equals (3, 4) is drawn from the origin. A line through the origin spans the direction u equals (1, 1). The projection of x onto that line lands at the point (3.5, 3.5), and the dashed residual from (3.5, 3.5) back to (3, 4) equals (minus 0.5, 0.5), meeting the line at a right angle. The angle between x and u is 8.13 degrees. span{u}, u = (1,1) x = (3, 4) proj = (3.5, 3.5) residual (−0.5, 0.5) ⟂ u θ = 8.13° 0 x₁ x₂
Notice that the projection is the unique point on the line whose residual is perpendicular to it, that orthogonality is the optimality condition, not a coincidence of the picture.

With \(x = (3,4)\) and \(u = (1,1)\): \(\langle x,u\rangle = 7\), \(\lVert u\rVert^2 = 2\), so \(\alpha^\star = 3.5\) and the projection is \((3.5, 3.5)\). The residual \((-0.5, 0.5)\) has inner product \(-0.5 + 0.5 = 0\) with \(u\), as promised. The angle satisfies \(\cos\theta = 7/(5\sqrt{2}) = 0.98995\), so \(\theta = 8.13^\circ\).

Orthonormal bases and Gram–Schmidt

A basis \(\{q_1,\dots,q_n\}\) is orthonormal when \(\langle q_i,q_j\rangle = \delta_{ij}\). It is the convenient case: coordinates are just inner products, \(x = \sum_i \langle x, q_i\rangle q_i\), and lengths are preserved by Parseval, \(\lVert x\rVert^2 = \sum_i \langle x,q_i\rangle^2\). Gram–Schmidt manufactures one from any basis \(\{a_1,\dots,a_n\}\) by repeatedly subtracting off what is already explained:

\[ v_k = a_k - \sum_{j<k} \langle a_k, q_j\rangle\, q_j, \qquad q_k = \frac{v_k}{\lVert v_k\rVert}. \]

Each step is the projection formula applied and subtracted. In floating point, classical Gram–Schmidt loses orthogonality badly on ill-conditioned inputs; the modified variant, or a Householder QR, is what numerical libraries use.

Hyperplanes and the margin

A hyperplane is the set \(H = \{x : \mathbf{w}^\top x + b = 0\}\) with \(\mathbf{w} \neq 0\). It is an \((n{-}1)\)-dimensional affine subspace, and \(\mathbf{w}\) is normal to it: for any \(x_1, x_2 \in H\), \(\mathbf{w}^\top(x_1 - x_2) = 0\). To find the distance from an arbitrary \(x_0\), step from \(x_0\) along \(\mathbf{w}\) until you hit \(H\). Write the foot as \(x_0 - t\mathbf{w}\) and solve \(\mathbf{w}^\top(x_0 - t\mathbf{w}) + b = 0\), giving \(t = (\mathbf{w}^\top x_0 + b)/\lVert \mathbf{w}\rVert^2\). The distance is \(\lVert t\mathbf{w}\rVert\):

\[ d(x_0, H) = \frac{\lvert \mathbf{w}^\top x_0 + b\rvert}{\lVert \mathbf{w}\rVert}. \]

Take \(\mathbf{w} = (2,1)\), \(b = -4\) and \(x_0 = (3,3)\): the numerator is \(\lvert 6 + 3 - 4\rvert = 5\) and \(\lVert\mathbf{w}\rVert = \sqrt 5\), so the distance is exactly \(\sqrt 5 \approx 2.236\).

Key Insight, where the SVM's \(2/\lVert \mathbf{w}\rVert\) comes from: the pair \((\mathbf{w}, b)\) is over-parameterised, since scaling both by \(c>0\) leaves the hyperplane unchanged. The hard-margin SVM removes that freedom by fixing the scale with the constraints \(y_i(\mathbf{w}^\top x_i + b)\ge 1\), so the closest points sit at \(\lvert \mathbf{w}^\top x + b\rvert = 1\). By the distance formula they are \(1/\lVert\mathbf{w}\rVert\) away, the two margins are \(2/\lVert\mathbf{w}\rVert\) apart, and maximising the margin becomes minimising \(\lVert\mathbf{w}\rVert^2\). The quadratic objective is not a modelling choice, it is the distance formula in disguise.

What this means for a trained classifier

The last layer of a neural classifier is a set of hyperplanes, one per class in the binary-versus-rest reading. The logit \(\mathbf{w}_c^\top h + b_c\) is a signed, unnormalised distance from the penultimate feature \(h\) to class \(c\)’s boundary; dividing by \(\lVert \mathbf{w}_c\rVert\) turns it into an actual distance. This is why logit magnitudes are not comparable across classes with differently scaled weight vectors, and why weight normalisation and cosine-softmax variants exist at all.

Interview trap: the formula \(\lvert\mathbf{w}^\top x_0 + b\rvert/\lVert\mathbf{w}\rVert\) is a distance only in the norm induced by the inner product you used. Under a different metric, a Mahalanobis inner product \(\langle x,y\rangle_A = x^\top A y\) with \(A \succ 0\), say, orthogonality, angles and the closest point on \(H\) all change. "Perpendicular" is a statement about a chosen inner product, never about the coordinates on the page.

Recap

  • Norm, angle and orthogonality are all defined from the inner product; change the inner product and all three change.
  • \(\mathrm{proj}_u(x) = \frac{\langle x,u\rangle}{\lVert u\rVert^2}u\), derived by minimising \(\lVert x - \alpha u\rVert^2\); the residual is orthogonal to \(u\) by construction. Least squares is the same statement in matrix form.
  • Gram–Schmidt is repeated projection-and-subtract; use the modified version or QR in floating point.
  • Distance from \(x_0\) to \(\{\mathbf{w}^\top x + b = 0\}\) is \(\lvert\mathbf{w}^\top x_0 + b\rvert/\lVert\mathbf{w}\rVert\). Fixing the scale with \(y_i(\mathbf{w}^\top x_i + b)\ge 1\) turns margin maximisation into minimising \(\lVert\mathbf{w}\rVert^2\).

Next: what happens when you move these vectors around, see linear transformations.

References

  1. Boyd, S., & Vandenberghe, L. Convex Optimization. Cambridge University Press, 2004.
  2. Cortes, C., & Vapnik, V. Support-Vector Networks. Machine Learning, 20(3), 1995.
  3. Trefethen, L. N., & Bau, D. Numerical Linear Algebra. SIAM, 1997.

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