Least square fitting of vector-valued random variables

Let (x, y) \in \mathbb{R}^n \times \mathbb{R}^m be joint random variables (typically not independent). This post describes which linear map \sigma{: \mathbb{R}^n \to \mathbb{R}^m} relates these variables best in the sense that the expected square error \mathbb{E} \lVert \sigma(x) - y \rVert^2 is minimised. Related applications such as principle component analysis and auto encoding under random distortion (such as drop out) are recovered as special cases.

Vectors will be interpreted as column vectors and transposition is denoted by a superscript asterisk. It is assumed that  X = \mathbb{E}(x x^{\ast}) is non-singular (and therefore positive definite). Define Y = \mathbb{E}(y x^{\ast}). Then Y X^{-1} Y^{\ast} is positive semi-definite on \mathbb{R}^m. Let v_1, \ldots, v_m \in \mathbb{R}^m be orthogonal eigenvectors in order of decreasing eigenvalue and let \pi_k be the orthogonal projection of \mathbb{R}^m onto the span of v_1, \ldots, v_k for k \in \{1, \ldots, m\}. (Note that this definition leaves some choice in case not all eigenvalues are distinct since the projections are not unique in that case.) Now the main result states:

For each k \in \{1, \ldots, m\} the linear map \sigma_k = \pi_k Y X^{-1} minimises the expected square error \mathbb{E} \lVert \sigma_k(x) - y \lVert^2 among all linear maps of rank k.

Let’s apply this result to two special cases. For the first case we simply assume that n = m, x=y and \mathbb{E}(x) = 0. In this case Y = X and Y X^{-1} Y^{\ast} = X. So the projections \pi_k project onto the eigenspaces of the covariance matrix X and \sigma_k = \pi_k Y X^{-1} = \pi_k. This result coincides with principle component analysis for the variable x.

For the second case assume that n = m and x = A y for some random diagonal matrix where each diagonal entry A_{ii} is independent (also of y) and Bernoulli distributed with probability p > 0. The matrix A models dropout in the coefficients of y. Let \Sigma = \mathbb{E}(y y^{\ast}) be the covariance matrix of y, D the diagonal of \Sigma, and q = 1-p. In this case

X = p \left( p \Sigma + q D \right) and Y = p \Sigma.

Now \pi_k are projections onto eigenspaces of

\Sigma \left(p \Sigma + q D \right)^{-1} \Sigma,

which is the semi-positive definite matrix that appeared in the previous post about linear encoders with dropout. Finally in this case \sigma_k = \pi_k \Sigma \left(p \Sigma + q D \right)^{-1}.

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Linear autoencoder with dropout

We begin with a brief review of least squares fitting formulated in autoencoder language. Let x be a random variable in \mathbb{R}^n such that \mathbb{E}(x) = 0 and let X = \mathbb{E}(x x^{\ast}) be its covariance matrix. Then X is a self-adjoint (symmetric) operator. Let d be a positive number not greater than n. A linear autoencoder for x is a pair of linear operators R{:\mathbb{R}^n \to \mathbb{R}^d} and C{: \mathbb{R}^d \to \mathbb{R}^n} such that

  1. The operator C is unitary: C^{\ast}C = 1 on \mathbb{R}^d.
  2. The expected square error \mathbb{E} \lVert x - CRx \rVert^2 is minimal among pairs (C, R).

The second requirement is interpreted as “R reduces the dimension of the variable x from n to d with a minimal loss of information”. The function

\displaystyle (C, R) \mapsto \mathbb{E} \lVert x - CRx \rVert^2

has a saddle point exactly when

  1. The image of C in \mathbb{R}^n is invariant under X.
  2. R = C^{\ast}.

Note that CR = C C^{\ast} is an orthogonal projection in this case. The expected error among such saddle points  is minimal if the image of C is a direct sum of eigenspaces of X with the largest possible eigenvalues.

The variable x is in practical situation a finite sampled set of observations x_1, x_2, \dots, x_m each occurring with equal probability. Doing a least square error fit on x as above has the risk of being oversensitive to features that are only apparent in this specific sample. In other words it may be sensitive to outliers. One way to reduce this sensitivity is to introduce dropout.

A standard form of dropout is the following. Let b = b(p) \in \{0, 1\} be a Bernoulli random variable with expectation p > 0 and B {: \mathbb{R}^n \to \mathbb{R}^d} a random operator with mutually independent coefficients b (so B_{ij} = e_i^{\ast} B e_j = b for all indices i, j). This matrix B is also taken to be independent of x. Now in dropout the operator R is replaced by the Hadamard product R \circ B. This means that each coefficient of R can “drop out” independently with probability q = 1-p. A linear autoencoder with dropout is a pair (C, R) of operators similar as above but now minimises the altered expected error

\displaystyle \mathbb{E}\left \lVert x - C (R \circ B) x \right \rVert^2.

Here the expectation is for the joint distribution of the independent pair (B, x). The idea is that R must now be robust against random dropout and that this prevents it from being oversensitive to accidental features in x. Also in this dropout case the saddle points of the function

\displaystyle (C, R) \mapsto \mathbb{E}\left \lVert x - C (R \circ B) x \right \rVert^2

can be described explicitly. Let \mathrm{diag}(X) denote the diagonal operator with the same diagonal entries as X. The pair (C, R) is a saddle point if

  1. The image of C is invariant under X \left( p X + q \, \mathrm{diag}(X) \right)^{-1} X.
  2. R  = C^{\ast} X \left(p X + q \, \mathrm{diag}(X)\right)^{-1}.

Indeed for p=1 (probability of dropout is zero) this reduces to the criterion above for a linear autoencoder without dropout.

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Mutual information and fast image registration

In the previous post the mutual information for a pair of images

\displaystyle{f = (f_1, f_2) {:[0,1]^2} \to [0,1]^2}

was expressed in terms of the angle \alpha between the gradients of f_1 and f_2:

\displaystyle{I(f) = \int_{[0,1]^2} -\tfrac12 \log \left(\sin^2(\alpha) \right) \mathrm{d}V = \int_{[0,1]^2} -\tfrac12 \log\left(\frac{1-\cos(2\alpha)}2\right) \mathrm{d}V}.

Note that if both images are equal then \alpha=0 everywhere and mutual information is not well defined in the form above. This can be fixed  by adding a Gaussian error term to the Jacobian of f as we will explore at the end of this post. Here we take a different approach that circumvents this problem altogether.

The goal of image registration is to position the reference image f_1 by shifting it relative to f_2 such that mutual information is maximized. At that  position the reference image f_1 provides the most information about f_2. This makes perfect sense for image registration. Note however that the formula above is rather expensive to use directly for this purpose since it has to be computed for every possible offset of the reference image f_1. One way around this is to apply some algorithm to reduce the number of positions to inspect such as steepest ascend. This is not what we will do here. Instead we use an approximation for mutual information that is efficient to compute for all positions.

Observing that the integrand p(\alpha) in the mutual information expression is a function of \cos(2\alpha) we will approximate it by trigonometric polynomials of the form

\displaystyle{p_d(\alpha) = \log(2) + \sum_{k=1}^dc_k \cos(2k \alpha)}

for some finite degree d \geq 1. The constant term is chosen such that

\displaystyle{\int_{-\pi/2}^{\pi/2}p(\alpha) \mathrm{d}\alpha = \int_{-\pi/2}^{\pi/2} p_d(\alpha) \mathrm{d}\alpha = \pi \log(2)}

for each trigonometric polynomial, so the area below all these functions is the same on the interval [-\pi/2, \pi/2]. The chosen approximation method considers these functions as probability densities. In particular each approximation p_d is chosen such that

  1. p_d(\alpha) \geq 0 for all \alpha
  2. p_d(-\pi/2) = p_d(\pi/2) = 0 and
  3. p_d minimizes the Kullback-Leibler divergence \mathrm{KL}(p, p_d) among all such trigonometric polynomials of degree d.

The first three approximations are:

  1. p_1(\alpha) \approx 0.693 + 0.693\cos(2\alpha)
  2. p_2(\alpha) \approx 0.693+0.912 \cos(2 \alpha)+0.219 \cos(4 \alpha)
  3. p_3(\alpha) \approx 0.693+0.969 \cos(2 \alpha)+0.425 \cos(4 \alpha)+0.149 \cos(6 \alpha)

Their graphs are shown below. Note that the peak around \alpha=0 gets more pronounced with an increasing degree. Also p_2 turns out to be very close (up to a scalar multiple) to \cos^4(\alpha) which is easier to remember and could be used as an alternative approximation.

image

Here is a plot of p_3 together with the actual distribution p.

image

Instead of the actual distribution p for mutual information we can use any of the approximations p_d to compute the approximate mutual information

\displaystyle{I_d(f) = \int_{[0,1]^2} p_d(\alpha) \, \mathrm{d}V}.

Moreover this value can be computed efficiently for every image offset of the reference image f_1 as a cross-correlation of  vector fields as we will see now. Identify \mathbb{R}^2 with the complex plane \mathbb{C}. The inner product on \mathbb{R}^2 translates to complex numbers as

\displaystyle{\langle z, w \rangle = \mathrm{Re}(z \overline{w})}.

Write the gradients of f_1 and f_2 normalized to length one as complex fields z_1 and z_2 (so \lvert z_1 \rvert = \lvert z_2 \rvert = 1). Then for the angle \alpha between these gradients and any integer k we have

\displaystyle{\cos(k\alpha) =  \langle z_1^k, z_2^k \rangle}.

This shows that every term in I_d(f) is a cross-correlation between two fields (z_1^k and z_2^k) and hence can be computed efficiently using FFT. Since for the reference image everything can be pre-computed the approximate mutual information I_d(f) can be computed with 2d+1 real FFT computations: two per term \cos(2k \, \alpha) for k=1 \ldots d to compute its spectrum and one reverse transform to compute the cross-correlation from a linear combination of all these spectra.

As a bonus this method can be used without any modification to locate only a part of the reference image f_1 in f_2: Simply mask out the part of f_1 to be ignored by setting its normalized gradient field z_1 to zero there. This works because the fields are normalized and are not affected by local variations in gain and offset when the masked reference image is matched with f_2. (In contrast, such variations make this simple masking scheme useless for an ordinary cross-correlatin approach.)

A final remark about how, as mentioned in the beginning of this post, error terms can be introduced in mutual information to get rid of the \log singularity at \alpha = 0. If the Jacobian appearing in the definition of mutual information is distorted by adding a Gaussian error term then mutual information is changed to

\displaystyle{I(t; f) = \int_{[0,1]^2} -\tfrac12 \log(\sin^2(\alpha) + t^2 \cos^2(\alpha)) \, \mathrm{d}V}

where t \in [0, 1] is an error ratio. Note that for t=0 we regain the original mutual information while for the other extreme t=1 we have I(1; f) = 0. Note that for t > 0 the singularity at \alpha=0 is resolved. For those who want to explore further in this direction I leave you with the remark that

\displaystyle{\int_{-\pi/2}^{\pi/2} -\tfrac12 \log(\sin^2(\alpha) + t^2 \cos^2(\alpha)) \, \mathrm{d}\alpha = -\pi \log \left( \frac{t+1}2 \right)}

which helps in treating the integrand as a probability density in the angle \alpha.

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Mutual information and gradients

In this post we will see a that mutual information between functions (e.g. two images) can be expressed in terms of their gradient fields. First some definitions and background. Let f{: [0, 1]^n} \to [0, 1]^m be a differential function for some n \geq m. In this post f typically represents an image or a pair of images with n=2 and m \in \{1, 2\}. Let D be the Jacobian of f. This is the m \times n matrix with

\displaystyle{D_{ij}=\frac{\partial f_i}{\partial x_j}} for 1 \leq i \leq m and 1 \leq j \leq n.

The entropy H(f) of f is defined by the integral

H(f) = \displaystyle{\int_{[0,1]^n} \tfrac12 \log \lvert D D^{\mathrm{t}}\rvert \, \mathrm{d}V}

where \lvert \cdot \rvert is the determinant and \mathrm{d}V the standard volume element. (In this post I will disregard any question of well-definedness of this integral.) To motivate  this definition: if n=m and f is injective then H(f) is the usual differential entropy of the push forward f_{\ast}(\mathrm{d}V) of the standard volume form.

If m=1 then the Jacobian D = \nabla f equals the gradient of f and \lvert D D^{\mathrm{t}} \rvert = \lVert \nabla f \rVert^2 so the entropy becomes

\displaystyle{H(f) = \int_{[0,1]^n} \log \lVert \nabla f \rVert \, \mathrm{d}V}.

Let f = (f_1, \ldots, f_m) then the mutual information I(f) of f is defined by

\displaystyle{I(f) = \sum_{k=1}^m H(f_k) - H(f)}.

Mutual information is always non-negative. It expresses how much information is gained by knowing the joint value distribution of f  compared to knowing only the value distributions of the separate coordinates f_1, \ldots, f_m. In other words, mutual information is a measure of dependence between the coordinates: The higher the dependence the higher the mutual information while for independent coordinates the mutual information is 0 (there is no information to be gained from their joint value distribution).

The nice thing about mutual information is that it is invariant under any injective coordinate-wise distortion. In imaging related terms it is for example invariant under changes of gamma, gain and offset of the image. This is hugely beneficial in practical imaging applications where lighting conditions are never the same. Different images (the coordinates) may even have been produced with completely different sensing equipment.

A key observation about mutual information is the following:

\displaystyle{\lvert D D^{\mathrm{t}} \rvert = v \cdot \prod_{k=1}^m \lVert \nabla f_k \rVert^2}

for some function v with values in  [0, 1] that depends only on the direction of the gradients but not their length. Moreover v=0 if and only if the gradients are linearly dependent and v=1 if and only if they are mutually orthogonal. Using this decomposition mutual information can be expressed as

\displaystyle{I(f) = \int_{[0,1]^n} -\tfrac12 \log(v) \, \mathrm{d}V}.

This confirms that mutual information is non-negative since v\in[0,1] and therefore \log(v) \leq 0. I will conclude this post by looking at the specific case of a pair of 2-dimensional images so the case that n=m=2. Then the function v has a simple explicit form. Let \alpha be the angle between the gradients \nabla f_1 and \nabla f_2. Then

v = \sin^2(\alpha) = \frac{1-\cos(2\alpha)}{2}.

There are two pleasant observations to make:

  1. Mutual information of a pair of images depends only on the double angle between their gradients. In particular it does not depend on the length or a sign change of either gradient.
  2. The expression \cos(2\alpha) is easy to compute as an inner product. The double angle can be accounted for by a simple rational transformation of the gradient. This will be explained in more detail in a next post.

A next post will discuss the application of mutual information to image registration. It results in a method that is very efficient (based on FFT), is robust against image distortions and can also be applied to register (locate) a partial template image of any shape within a bigger scene.

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Projections of the Mahalanobis norm

In this post \langle \cdot,\cdot \rangle is the standard inner product on \mathbb{R}^n for some n \geq 2 and A is a fixed Hermitian positive definite operator. The square Mahalanobis norm m of a vector v \in \mathbb{R}^n for this operator is defined by

\displaystyle m(v) = \langle v, A^{-1}v \rangle.

The results in this post were found while looking for ways to approximate this Mahalanobis norm without the need to invert A. (Later I realised that using the Cholesky factorisation suited me better. Nice results can be found by looking in the wrong places!) The idea is to use projections on some smaller dimensional subspace to get estimates of the actual Mahalanobis norm. To be precise let V \subseteq \mathbb{R}^n be some subspace of dimension 1 \leq d \leq n and let \pi_V be the orthogonal projection onto V. The operator \pi_V A is non-singular on the subspace V. Let A_V^{-1}:\mathbb{R}^n\to V be its pseudo inverse such that A_V^{-1}A\pi_V = \pi_V A A_V^{-1} = \pi_V. The projected Mahalanobis norm m_V on V is defined by

\displaystyle m_V(v) = \langle A_V^{-1} v, v \rangle.

Let’s take the one-dimensional case as an example. Let v \in \mathbb{R}^n be non-zero and denote the span of v by \llbracket v \rrbracket. Then the norm m_{\llbracket v \rrbracket} is given by

\displaystyle m_{\llbracket v \rrbracket}(v) = \frac{\langle v,v \rangle^2}{\langle A v, v \rangle}.

Note that this expression does not involve the inverse of A. The basic property of the projected Mahalanobis norm is the following:

The inequality m_V \leq m holds throughout V. Equality m_V(v) = m(v) occurs if and only if A_V^{-1}v = A^{-1}v.

This property follows from the Cauchy-Schwarz inequality for the inner product \langle \cdot, A^{-1} \cdot \rangle:

\displaystyle \langle A_V^{-1}v, v\rangle^2 = \langle A A_V^{-1}v, A^{-1} v\rangle^2 \leq \langle A A_V^{-1}v, A_V^{-1} v \rangle \langle v, A^{-1}v\rangle = \langle v, A_V^{-1} v \rangle \langle v, A^{-1}v\rangle.

This is an equality if and only if AA_V^{-1}v and v are linearly dependent. Combined with \pi_VAA_V^{-1}v = v it follows that in fact AA_V^{-1}v = v.

The following realisation came as a surprise. It shows that projections onto two-dimensional subspaces suffice to get an exact value for the Mahalanobis norm:

Let v\in \mathbb{R}^n be a non-zero vector and let V \subseteq \mathbb{R}^n be the span of \{v, A^{-1}v \} (so \dim(V) \leq 2). Then m_V(v) = m(v).

The projected norm for a two-dimensional subspace also has a simple explicit form. Let w \in \mathbb{R}^n be a non-zero vector orthogonal to v and let V be the span of \{v, w\}. The norm m_V is given by

\displaystyle m_V(v) = \frac{\langle v, v \rangle^2}{\langle A v, v\rangle - \frac{\langle A v, w \rangle^2}{\langle A w, w\rangle}}.

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The quantum harmonic oscillator in the plane

This post investigates some solutions of the (or rather a) quantum harmonic oscillator in the plane. Here the plane is \mathbb{C} which will be identified with \mathbb{R}^2 through z=x+\textrm{i} y. Let \partial_x and \partial_y denote the partial derivative operators with respect to x and y respectively. Then the Laplace operator \nabla^2 is \partial_x^2 + \partial_y^2. The time independent form of the Schrödinger equation for the quantum harmonic oscillator considered here is

\displaystyle (-\tfrac{1}{4} \nabla^2 + |z|^2) \varphi = E \varphi

where \varphi is a complex function and E a real eigenvalue (related to energy in physical terms). The operator -\tfrac{1}{4} \nabla^2 + |z|^2 is called the Hamiltonian and will be denoted by H. Only radial solutions \varphi will be considered. A complex function is called radial when it is of the form

\displaystyle \varphi(z) = f(|z|^2) z^m or \displaystyle \varphi(z) = f(|z|^2) \overline{z}^{\,m}

for some real analytic function f and some m \in \mathbb{N}. Radial functions of the first and second form are said to be of degree m and -m respectively. If \varphi is a radial function of degree m \in \mathbb{Z} and |\omega| = 1 then \varphi(\omega z) = \omega^m \varphi(z).

Working over \mathbb{C} it is convenient to rewrite the Laplace operator as follows. Let \partial = \tfrac{1}{2}(\partial_x - \textrm{i}\partial_y) and \overline{\partial} = \tfrac{1}{2}(\partial_x + \textrm{i} \partial_y). If f is a complex differentiable function then \overline{\partial} f = 0 and \partial f = f' while \overline{\partial}\, \overline{f} = \overline{f'} and \partial \overline{f} = 0. (These are the Cauchy-Riemann equations.) The operators \partial and \overline{\partial} commute and are related to the Laplace operator by

\displaystyle \partial \overline{\partial} = \overline{\partial} \partial = \tfrac{1}{4} \nabla^2.

The Hamiltonian is therefore -\partial \overline{\partial} + |z|^2 and we will use it in this form. For a radial function f(|z|^2) z^m

\displaystyle \partial \overline{\partial} f(|z|^2) z^m = \partial f'(|z|^2) z^{m+1} = |z|^2 f''(|z|^2) z^m + (m+1) f'(|z|^2) z^m.

Applying the Hamiltonian results in

H f(|z|^2) z^m = |z|^2 \left(f(|z|^2) - f''(|z|^2)\right) z^m - (m+1) f'(|z|^2) z^m

and in particular if f(x) = e^{-x} then H e^{-|z|^2} z^m = (m+1) e^{-|z|^2} z^m. So for each degree m \geq 0 the function e^{-|z|^2} z^m is an eigenfunction of H with eigenvalue E = m + 1. Complex conjugation shows that the same holds for e^{-|z|^2} \overline{z}^{\,m} of degree -m. For each degree m \in \mathbb{Z} we found a radial solution for the harmonic oscillator of degree m and eigenvalue |m| + 1. These examples do not exhaust all such solutions however. Others can be found by a clever trick that was already known in this context by Schrödinger and Dirac.

This trick is known as the factorisation or algebraic method and the auxiliary operators that appear are called ladder operators or annihilation and creation operators. The following facts can be readily verified:

  1. The operators -\partial+\overline{z} and \partial + \overline{z} lower the degree of a radial function by 1.
  2. The operators -\overline{\partial} + z and \overline{\partial} + z raise the degree of a radial function by 1.
  3. If \varphi is a radial function of degree m \in \mathbb{Z} then (z \partial - \overline{z} \, \overline{\partial}) \varphi = m \varphi and

    \displaystyle (-\partial + \overline{z})(\overline{\partial} + z) \varphi = (H - 1 - m) \varphi
    \displaystyle (\overline{\partial} + z)(-\partial + \overline{z}) \varphi = (H + 1 - m) \varphi
    \displaystyle (-\overline{\partial} + z)(\partial + \overline{z}) \varphi = (H - 1 + m) \varphi
    \displaystyle (\partial + \overline{z})(-\overline{\partial} + z) \varphi = (H + 1 + m) \varphi

Combining these observations we find for a radial function \varphi of degree m:

\displaystyle \begin{matrix}  H(-\partial + \overline{z})\varphi &=& ((-\partial + \overline{z})(\overline{\partial} + z) + m)(-\partial + \overline{z})\varphi\\   &=& (-\partial + \overline{z})((\overline{\partial} + z)(-\partial + \overline{z}) + m)\varphi\\   &=&(-\partial + \overline{z})(H + 1)\varphi.  \end{matrix}

In particular if \varphi \neq 0 is a solution with eigenvalue E then (-\partial + \overline{z})\varphi is a solution of degree m-1 and eigenvalue E+1. In this case (\overline{\partial} + z)(-\partial + \overline{z})\varphi = (E + 1 - m) \varphi and so (-\partial + \overline{z})\varphi \neq 0 if E \neq m - 1. Under the assumption that E \geq |m| + 1 we can assemble the following table of solutions based on \varphi:

\displaystyle \begin{matrix}  \textrm{solution} & \textrm{degree} & \textrm{eigenvalue} & \textrm{remark}\\[1ex]  (-\partial + \overline{z})\varphi & m - 1 & E + 1 & \neq 0\\  (-\overline{\partial} + z)\varphi & m + 1 & E + 1 & \neq 0\\  (\partial + \overline{z})\varphi & m - 1 & E - 1 & \neq 0 \textrm{ unless } E = 1 - m\\  (\overline{\partial} + z)\varphi & m + 1 & E - 1 & \neq 0 \textrm{ unless } E = 1 + m  \end{matrix}

The first two operators in this table are called raising or creation operators: they raise the eigenvalue or create energy. The last two are called lowering or annihilation operators for similar reasons. Starting from the solution \varphi(z) = e^{-|z|^2} of degree m=0 and eigenvalue E=1 and repeatedly applying the operators -\partial + \overline{z} and -\overline{\partial} + z we find non-zero solutions for all pairs (m, E) for which m + E is odd and E \geq |m|+1. This process results in the solutions below up to scalar multiples. Solutions for negative degrees can be obtained by complex conjugation.

\displaystyle \begin{matrix}  m & E & \textrm{solution}\\[1ex]  0 & 1 & e^{-|z|^2} \\  1 & 2 & z \, e^{-|z|^2}\\  0 & 3 & (1 - 2|z|^2) \, e^{-|z|^2}\\  2 & 3 & z^2 \, e^{-|z|^2}\\  1 & 4 & z (1 - |z|^2) \, e^{-|z|^2}\\  3 & 4 & z^3 \, e^{-|z|^2}  \end{matrix}

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Bounds for central binomial coefficients

Let n be a positive integer. The binomial coefficient {2n \choose n} is called a central binomial coefficient. Several bounds for these coefficients are known and the more advanced ones are commonly derived from Stirling’s formula for factorials or from Wallis’ product for \pi. This post presents an alternative and self-contained elementary proof of the bounds

\displaystyle \frac{4^n}{\sqrt{(n+\frac{1}{2})\pi}} < {2n \choose n} < \frac{4^n}{\sqrt{n\pi}}.

Define a function f by

\displaystyle f(x) = \cos(x)^n e^{n x^2/2}

and take x\in[0, \pi/2). From f(x) \geq 0 and \tan(x)\geq x it follows that

\displaystyle f'(x) = n f(x)\left(x - \tan(x)\right) \leq 0.

So f is non-increasing on this interval and therefore f(x) \leq f(0)=1 or

\displaystyle \cos(x)^n \leq e^{-n x^2/2}.

This inequality clearly also holds at the endpoint x=\pi/2 and since both sides are symmetric in x it holds throughout the interval [-\pi/2, \pi/2]. (It suggests that \cos(x)^n closely resembles a normal distribution on this interval.) Integration of the inequality above leads to

\displaystyle \int_{-\pi/2}^{\pi/2}\cos(x)^n dx \leq \int_{-\pi/2}^{\pi/2}e^{-n x^2/2}dx < \int_{-\infty}^{\infty}e^{-n x^2/2}dx.

Both the far left and right hand sides of these inequality can be computed explicitly. Starting with the right hand side let

\displaystyle b_n=\int_{-\infty}^{\infty}e^{-n x^2/2}dx.

Then

\displaystyle b_n^2=\int_{-\infty}^{\infty}e^{-n x^2/2}dx \int_{-\infty}^{\infty}e^{-n y^2/2}dy=\iint_{-\infty}^{\infty}e^{-n (x^2+y^2)/2}dx dy

\displaystyle =2 \pi \int_0^{\infty} r e^{-n r^2/2}dr = \frac{2\pi}{n}

And so b_n=\sqrt{\frac{2\pi}{n}}. To evaluate the left hand side let

\displaystyle a_n=\int_{-\pi/2}^{\pi/2}\cos(x)^n dx.

By explicit computation we find a_0=\pi and a_1=2. The other values can be found by a recursive relation that follows from partial integration. For positive n we have

\displaystyle a_{n+1}=\int_{-\pi/2}^{\pi/2}\cos(x)^n \cos(x)dx=n\int_{-\pi/2}^{\pi/2}\cos(x)^{n-1} \sin(x)^2dx

\displaystyle =n\int_{-\pi/2}^{\pi/2}\cos(x)^{n-1}\left(1- \cos(x)^2\right)dx=n(a_{n-1}-a_{n+1})

and therefore the recursion a_{n+1}=\frac{n}{n+1}a_{n-1}. Using this recursion one can check that the even and odd entries of the sequence a are given respectively by

\displaystyle a_{2n}=4^{-n}{2n \choose n}\pi

\displaystyle a_{2n+1}=\frac{2^{2n+1}}{{2n \choose n}(2n +1)}.

Putting all results sofar together we find for positive even integers 2n

\displaystyle 4^{-n}{2n \choose n}\pi<\sqrt{\frac{2\pi}{2n}}

\displaystyle {2n \choose n}<\frac{4^n}{\sqrt{n \pi}}

and for odd integers 2n+1

\displaystyle \frac{2^{2n+1}}{{2n \choose n}(2n+1)} < \sqrt{\frac{2\pi}{2n+1}}

\displaystyle \frac{4^n}{\sqrt{(n+\frac{1}{2})\pi}} < {2n \choose n}.

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Fixed points of entire functions

This story has a long personal history. In my first attempted complex analysis exam I was asked if the entire function e^z has any fixed points. I had no idea how to approach this question and I failed the exam. This first failure was soon overcome but that particular question was —much to my advantage— never addressed during my study again. Much later I found several ways to solve this question.

The image of e^z-z is invariant under a shift z \mapsto z + 2 \pi \textrm{i}. The “Little Picard” theorem then asserts that this function maps onto \mathbb{C} since it cannot omit just a single value and so it must have a root. Another way uses “Great Picard” and the fact that z e^{-z} only has a single root so it must attain the value 1 infinitely often. Both approaches seem nice enough but depend on non-trivial theorems (“Great Picard” more so than “Little Picard”) and there is no indication of the location of the fixed points. A more pedestrian approach shows that there must be infinitely many fixed points along the curve |e^{2z}| = |z|^2 but this is hardly in the spirit of complex analysis. This was the state of affairs until this week. Then I found a simple and much more satisfying answer.

The Banach fixed point theorem asserts that every contraction of a complete metric space has a single fixed point. I will use that theorem in the following setting.

Let E be a non-empty closed convex subset of an open set U and let f{:U}\to E be holomorphic. If there exists a constant k < 1 such that |f'| \leq k on E then f restricted to E is a contraction and therefore f has a single fixed point in E.

For m \in \mathbb{Z} let E_m denote the horizontal strip

\displaystyle E_m = \{ z \mid |\textrm{Im}(z) - 2m \pi| \leq \pi \}

and let \log_m be the branch of the logarithm that maps the slit complex plane \mathbb{C} \setminus (-\infty, 0] onto the interior of E_m. If m \neq 0 then

\displaystyle |\log_m'| \leq \frac{1}{(2|m|-1)\pi} < 1

on E_m so the conditions of the theorem above apply and therefore \log_m has a single fixed point z_m \in E_m (which must lie in the interior.) Each z_m is a fixed point of e^z. Unfortunately the same argument does not work for E_0 where e^z has two more fixed points.

The fixed point method works equally well for a number of other functions. Here are two more examples:

Example 1. Let m \in \mathbb{Z} and m \neq 0. The branch of \arcsin that maps the (closed) upper half plane onto the half strip

\displaystyle E_m = \{ z \mid \textrm{Im}(z) \geq 0 \textrm { and } |\textrm{Re}(z) - 2m\pi| \leq \frac{\pi}{2} \}

is a contraction on E_m. So \sin(z) has exactly one fixed point in each such strip (and therefore also exactly one in its complex conjugate).

Example 2. Let m \in \mathbb{Z} and m odd. The branch of \arctan that maps the complement of the open unit disc into the strip

\displaystyle E_m = \{ z \mid |\textrm{Re}(z) - \tfrac{1}{2}m \pi| \leq \frac{\pi}{4} \}

is a contraction on E_m. So \tan(z) has exactly one fixed point in each such strip.

These example for e^z, \sin and \tan work so well because their inverses have a nice derivative that is less than one except in a small bounded region. Put in another way these functions all satisfy a differential equation of the form (f')^d = p(f) for some d \in  \{1,2\} and some polynomial p.

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Acceleration of alternating series

Let f(n):\mathbb{N}\to\mathbb{R} be a positive decreasing function such that \lim_{n\to\infty}f(n)=0. Then the alternating series

\displaystyle \sum_{n=0}^{\infty}(-1)^nf(n)

converges. Depending on the rate at which f decays it may however take many terms to get a decent approximation of the sum. Examples of such slowly converging series are

\displaystyle \log(2) = \sum_{n=0}^{\infty}(-1)^n\frac{1}{n+1}

\displaystyle \frac{\pi}{4} = \sum_{n=0}^{\infty}(-1)^n\frac{1}{2n+1}

A well known method to accelerate the convergence of such alternating series uses the binomial transform of the function f as follows. For m \geq 0 let \Delta^m f be the m-th difference of f:

\displaystyle (\Delta^m f)(n) = \sum_{k=0}^m (-1)^k{m \choose k} f(n+k)

Then it turns out that

\displaystyle \sum_{n=0}^{\infty}(-1)^nf(n) = \sum_{n=0}^{\infty}\frac{(\Delta^n f)(0)}{2^{n+1}}

and the right hand side may converge much faster. The binomial transform for the two examples above can be computed explicitly to obtain

\displaystyle \log(2)=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}(n+1)}

\displaystyle \frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{n!}{2\cdot(2n+1)!!} = \frac{1}{2} + \frac{1}{2\cdot 3} + \frac{2}{2 \cdot 3 \cdot 5} + \frac{2 \cdot 3}{2 \cdot 3 \cdot 5 \cdot 7} + \ldots

Both transformed series indeed converge much faster than the original ones. You may recognize the first as the series for -\log(1-x) taken at x=\tfrac{1}{2}. The second one is harder to spot but it turns out to be the series of

\displaystyle \frac{\arcsin(x)}{\sqrt{2-2x^2}}

taken at x=\tfrac{1}{2}\sqrt{2}.

The binomial transform is a great tool if you can find an explicit expression for it such as in both examples above. This is however not a trivial transform in general. If you do not know an explicit expression for the binomial transform then it is also not very convenient to compute: (\Delta^n f)(0) depends on the values f(k) for all k \leq n.

There is however another much simpler method that only involves the repeated difference \Delta^m f for a single fixed value of m \geq 1. This difference is now easy to compute (numerically) since (\Delta^m f)(n) depends only on m+1 consecutive values of f. It is convenient for the sake of notation to extend f to \mathbb{Z} by setting f(n)=0 for all n < 0. Then (\Delta^m f)(n) = 0 for all n < -m.

Now the following equalities hold in which the second and third sums are taken over \mathbb{Z}:

\displaystyle \sum_{n=0}^{\infty}(-1)^nf(n) = 2^{1-m}\sum_{n \textrm{ even}}(\Delta^m f)(n) = -2^{1-m} \sum_{n \textrm{ odd}} (\Delta^m f)(n)

If all terms (\Delta^m f)(n) are positive for n \geq 0 then partial sums increase for the second series and decrease for the third from n=0 onward. This means that these partial sums are lower and upper bounds for the series which is useful to estimate the remaining error for these partial sums.

As an example take m=2. The series for \log(2) and \tfrac{\pi}{4} now become

\displaystyle \log(2) = \frac{1}{2} +  \sum_{n=0}^{\infty}\frac{1}{(2n+1)(2n+2)(2n+3)} = \frac{3}{4} - \sum_{n=0}^{\infty}\frac{1}{(2n+2)(2n+3)(2n+4)}

\displaystyle \frac{\pi}{4} = \frac{1}{2} +  \sum_{n=0}^{\infty}\frac{4}{(4n+1)(4n+3)(4n+5)} = \frac{5}{6} - \sum_{n=0}^{\infty}\frac{4}{(4n+3)(4n+5)(4n+7)}

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Game trees under a partial order part 3: Alpha Beta pruning

In this last post of this series of three I will discuss alpha beta pruning. The context will be evaluation functions on game trees that take values in a distributive lattice. Of course a prime example would be the linear lattice \mathscr{L}(X, \leq) over the partially ordered set (X, \leq) but this post does not depend on specific features of this linear lattice. The result is a description of alpha beta pruning for distributive lattices that is a slightly improved version of that given by Ginsberg et al in their paper “Alpha Beta pruning under partial orders” from 2001.

Alpha Beta pruning is a way to compute the value of an evaluation function on a game tree for the root node by visiting as little child nodes as necessary. See the previous post for a more extensive introduction to evaluation functions. Here I will only briefly recall the recursive definition of an evaluation function f on a game tree that takes values in a lattice L. I will assume that L has a minimal and maximal element denoted by \mathbf{0} and \mathbf{1}. The normal forms \textrm{CNF}(X, \leq) and \textrm{DNF}(X, \leq) of the linear lattice are examples of this.

  1. The value of f for a leaf node is given by some heuristic game evaluation function (e.g. chess piece values).
  2. At an inner min node (where the minimizing player is to make a move) the value of f is defined recursively as the greatest lower bound in L of all values of f on direct child nodes.
  3. At an inner max node the value of f is defined as the least upper bound of all values of f on direct child nodes.

This definition leads to the following algorithm for the computation of f. This algorithm is also the basis of further considerations on ways to prune parts of the game tree that are irrelevant for the computation of f on the root node.

function f(node)
  local value
  if node is a leaf node then
    value = heuristic position value of node
  elseif node is a min node then
    value = 1
    for child in all children of node do
      value = value ∧ f(child)
    end
  else
    value = 0
    for child in all children of node do
      value = value ∨ f(child)
    end
  end
  return value
end

The structure of this naive algorithm is simple. To compute the value at a min node start with the maximal value \mathbf{1} then iterate over all child nodes and adjust the node’s value by taking the greatest lower bound with the value of a child node. At a max node the procedure is similar but starts off with the minimal value \mathbf{0} and repeatedly takes the least upper bound. The computation recurses depth first left to right over all child nodes, where “left to right” is according to the enumeration order of child nodes. Move ordering does not play a role in this naive algorithm but it can play a big role in alpha beta pruning in terms of the number of pruned nodes. I will not address move ordering in this post. During execution of the algorithm the state of the computation is schematically depicted below. As in the example in the previous post I will assume that the top node is a min node.

Intermediate result in the recursive computation of an evaluation function

Intermediate values in the depth first left to right computation of an evaluation function

Here x_1, x_2, \dotsc and y_1, y_2, \dotsc are the intermediate values that are computed for the inner min and max nodes respectively. The \star symbol indicates all subtrees located to the left of the current path of computation in the tree. These are the subtrees for which a value is already (recursively) computed. When the computation progresses each intermediate value x_j descends from \mathbf{1} and each y_j ascends from \mathbf{0}. When we would cut the computation short at the point depicted above then the algorithm would track back up and yield the final outcome

x_1 \wedge (y_1 \vee (x_2 \wedge \dotsc \vee (x_n \wedge y_n)))

at the root node. The essential point of alpha beta pruning is that it identifies when this intermediate value remains constant even when y_n gets larger in the remainder of the computation. Because in that case any further children of the node marked y_n in the diagram will not contribute to the final result and can be pruned (skipped). In the picture above the search path ends with a max node but that is irrelevant for alpha beta pruning as we will see.

A lattice L has a canonical partial order where x \leq y is equivalent to x \wedge y = x (or alternatively x \vee y = y). The lattice operators \wedge and \vee indeed result in the greatest lower bound and least upper bound with respect to this ordering. The following lattice property is the key point in alpha beta pruning.

Lemma: When x, y, z \in L and y \leq z then x \wedge y \leq x \wedge z and x \vee y \leq x \vee z.

Since y \leq z the greatest lower bound x \wedge y is also a lower bound for x and z and is therefore less than or equal to the greatest lower bound x \wedge z. The other inequality is derived similarly. For sequences x_1, x_2, \dotsc and y_1, y_2, \dotsc in L define sequences \alpha and \beta recursively by

  1. \alpha_1 = x_1 \wedge y_1 and \alpha_{j+1} is obtained from \alpha_j by the substitution y_j \leftarrow y_j \vee (x_{j+1} \wedge y_{j+1}).
  2. \beta_1 = x_1 and \beta_{j+1} is obtained from \beta_j by the substitution x_j \leftarrow x_j \wedge (y_j \vee x_{j+1}).

So the sequence \alpha starts with

\alpha_1 = x_1 \wedge y_1
\alpha_2 = x_1 \wedge (y_1 \vee (x_2 \wedge y_2))
\alpha_3 = x_1 \wedge (y_1 \vee (x_2 \wedge (y_2 \vee (x_3 \wedge y_3))))
\vdots

and the sequence \beta with

\beta_1 = x_1
\beta_2 = x_1 \wedge (y_1 \vee x_2)
\beta_3 = x_1 \wedge (y_1 \vee (x_2 \wedge (y_2 \vee x_3)))
\vdots

The following inequalities are a direct consequence of our lemma above. The sequences \alpha and \beta satisfy

\alpha_j \leq \alpha_{j+1} \leq \beta_{j+1} \leq \beta_j

for all indices j. In particular \alpha is an ascending sequence, \beta is a descending sequence, each \alpha_j is a lower bound for \beta and each \beta_j is an upper bound for \alpha. If \alpha_j = \beta_j or \alpha_j = \beta_{j+1} for some index j then both sequences are constant from that point onwards. The values in the sequences \alpha and \beta are precisely those obtained from aborting the computation of the evaluation function for the root node at min node j (sequence \alpha) or at max node j (sequence \beta). Hence further nodes can be pruned from the computation as soon as both sequences yield equal values.

Alpha Beta pruning (strong version): At the first index j where either \alpha_j = \beta_j or \alpha_j = \beta_{j+1} further subnodes of max node j or min node j+1 can be pruned.

Note that this formulation of alpha beta pruning resembles “classical” alpha beta pruning for totally ordered sets in that we have an ascending sequences of high values and a descending sequence of low values and nodes can be pruned as soon as these values coincide. There is also an unfortunate difference between the two. In classical alpha beta pruning the high and low values are easy to compute progressively as a simple maximum or minimum value. In contrast high and low values in the pruning algorithm above are not so simple to compute. Since they are computed deepest node first it requires constant back tracking. Moreover, computation of \wedge and \vee can be much more expensive than taking simply the maximum or minimum of two numbers.

Also note that the distributive property of the lattice L is not used at all in this version of alpha beta pruning! This does not contradict the results of Ginsberg et al since they formulate alpha beta pruning (in particular “deep pruning”) differently. I will briefly come back to this point at the end of this post.

There is a weaker version of alpha beta pruning based on alternative sequences \alpha' and \beta' that are easier to compute. The consequence is however that it may take longer before the ascending high values and descending low values allow cut off, resulting in less aggressive pruning. The weaker estimates for high and low values follow from the following identities (where the distributive property of L is now essential). Let y_0 = \mathbf{0} and define two new sequences \alpha' and \beta' by

\alpha'_j = y_0 \vee \dotsc \vee y_{j-1} and
\beta'_j = x_1 \wedge \dotsc \wedge x_j.

So \alpha'_j is the least upper bound of the values of all max nodes above min node j and \beta'_j is the greatest lower bound of the values of all min nodes above max node j. If we take \alpha_0 = \mathbf{0} and \beta_0 = \mathbf{1} then for all indices j \geq 1 the following equalities hold:

\alpha_j = \alpha_{j-1} \vee (\beta'_j \wedge y_j) and
\beta_j = \beta_{j-1} \wedge (\alpha'_j \vee x_j).

These can be proved by induction and the distributive property of the lattice L. Taking y_j = x_j in the first equation and x_j = y_{j-1} in the second results in

\beta_j = \alpha_{j-1} \vee \beta'_j and
\alpha_{j-1} = \beta_{j-1} \wedge \alpha'_j.

Comparing these equalities it follows that x_j \leq \alpha'_j implies \beta_j = \alpha_{j-1} and \beta'_j \leq y_j implies \alpha_j = \beta_j. This leads to the following version of alpha beta pruning.

Alpha Beta pruning: If the value x_j of min node j gets less than or equal to \alpha'_j or the value y_j of max node j gets greater than or equal to \beta'_j then further child nodes can be pruned.

This is a slightly different formulation of alpha beta pruning than Ginsberg et al use in their paper. What they call (shallow or deep) pruning comes down to either x_j \leq y_k for some k < j at min node j or x_k \leq y_j for some k \leq j at max node j. These are stronger conditions than those we found above and will therefore prune fewer nodes in general.

Here is an algorithmic formulation of alpha beta pruning. The value of the root node is computed by f(root, 0, 1).

function f(node, α, β)
  local value
  if node is a leaf node then
    value = heuristic position value of node
  elseif node is a min node then
    value = 1
    for child in all children of node do
      value = value ∧ f(child, α, β ∧ value)
      if value ≤ α then break end
    end
  else
    value = 0
    for child in all children of node do
      value = value ∨ f(child, α ∨ value, β)
      if value ≥ β then break end
    end
  end
  return value
end

To sum up the results of this post: I described two versions of alpha beta pruning that allow to prune nodes from the computation of an evaluation function with values in a distributive lattice at the root node. The first, strong, version has the disadvantage that it is computationally more difficult than the second. Both versions are stronger (potentially prune more nodes) than the method described in the paper by Ginsberg et al.

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