Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra I: Matricial Data

As in statistical physics, in the big data area, a state is a point in a space with a very large number of dimensions. PCA, in turn, aims to find the most relevant features out of a very large number of variables. In the case of a continuous spectrum, the relevance is fixed by some sensitivity threshold, discriminating between large eigenvalues that we keep and small eigenvalues that we discard. In other words, PCA constructs effective descriptions, which are valid as long as we can ignore the small eigenvalue effects. This picture is reminiscent of what RG do. From this, it could exist a criterion to distinguish between a noisy spectrum and another containing information, based on the differences between their respective effective asymptotic states. Let us consider the example of the field theory described by the classical action (3). The dimension of coupling constants like g depends on the dimension of the background space ${\mathbb{R}}^d$. In this example, $[g]=d-4$. The relevance in the vicinity of the Gaussian fixed point ($g\approx 0$) depends on the value of this dimension. For $d>4$, the operator ${\varphi }^4(x)$ is irrelevant, meaning that for sufficiently large scale, the theory is essentially Gaussian. In the opposite situation, for $d<4$, the RG flow moves away from the Gaussian fixed point (The point $m^2=g=0$ is a fixed point of the RG flow, thus any partial integration leads to another Gaussian model in virtue of the Gaussian integration properties. Thus, the relevance of the operators in the deep IR is usually determined by the dimension of the space.) However, the latter determines also the shape of the distribution for the Laplacian eigenvalues $p^2$, which is $\rho (p^2)={(p^2)}^{d/2-1}$, and relevance can be alternatively viewed as a property of the momentum distribution, without reference to the background space dimension. This is exactly what the authors of references [20,21,22,23] did: the scaling dimension is defined through the coarse graining, from the requirement that no explicit scale dependence occurs in the flow equations, excepts eventually at the linear level (see Section 3.1).

We propose a framework allowing to construct a field theoretical approximation of the fundamental RG flow. This point of view is familiar in condensed matter physics, and especially in the physics of critical phenomena. The classical example is the Ising model, whose critical behavior may be well approximated by an effective field theory in the critical domain [7]. Let us consider a set of N random variables, $\varphi =\{{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_N\}\in {\mathbb{R}}^N$; providing an archetypal example of field. Moreover, we assume that there exists a distribution $p[\varphi ]$ able to reproduce the covariance matrix $\mathcal{C}$, at least for sufficiently large scale (in eigenvalue space). An elementary formal realization of this is given by the Gaussian states: ensuring that $\langle {\varphi }_i{\varphi }_j\rangle ={\mathcal{C}}_{ij}$. The bracket notation $\langle X[\varphi ]\rangle$ defines the mean value of X with respect to the distribution $p[\varphi ]$. For such a Gaussian description, all the non-vanishing momenta of the distributions, $\langle {\varphi }_i{\varphi }_j{\varphi }_k\cdots \rangle$ reduce to a sum of the product of 2-points function following Wick theorem, and only the second cumulant does not vanish. In the field theory language, a theory with this property is said to be free. From an RG point of view, this description makes sense only if the Gaussian point is stable, i.e., if any perturbation around the Gaussian point ends up disappearing after some steps of the RG. In the next section we show that for MP law the Gaussian point is unstable. Moreover for a realistic dataset, correlations for n-point functions fail to be a product of 2-point functions, as a purely Gaussian law would have required. In the field theory language, this failing of the Gaussian description indicates the presence of interactions, materialized as non-Gaussian terms in the action. The coupling $g\int {\varphi }^4(x)$ in (3) provides an elementary example.

We, therefore, consider an interacting field theory. In standard field theory, there exist powerful principles, inherited from physics or mathematical consistency, to guide the choice of interactions, and the relevant domains of the theory space. In the absence of such a guide, we use the same simplicity argument already considered in [20,23]. We focus on purely local interactions of the form $g{\sum }_i{\varphi }_i^{2n}$, with fields interacting on the same point, with the same coupling constant. In that way, near to the Gaussian point our distribution $p[\varphi ]$ is suitably expanded as Note that, in our assumptions we kept only even interaction terms, ignoring for instance couplings like ${\varphi }_i^3$. This hypothesis is equivalent to assume the reflection symmetry (note that, truncating around quartic interactions, adding a term like ${\sum }_{i,j}{\varphi }_i^2{\varphi }_j^2$, which is invariant under to the rotational group $O(N)$, enlarges the discrete group ${\mathbb{Z}}_2$ to the hypercubic symmetry. Thus, the purely local model is, with this respect, the less structured one.) ${\varphi }_i\to -{\varphi }_i$. Finally, note that, in principle, ${\tilde{\mathcal{C}}}^{-1}\ne {\mathcal{C}}^{-1}$. Indeed, what is known “empirically” is the full 2-point function ${\mathcal{C}}_{ij}$; and the probability distribution must be: and from perturbation theory Note that, from an information theory point of view, probability density (5) corresponds to the maximum entropy solution, compatible with constraint (5) and the existence of undetermined non-Gaussian correlations. From that point of view, the model is minimal in the sense that it carries the least possible structure, as stated in [20]. In that setting ${\mathcal{C}}_{ij}={\tilde{\mathcal{C}}}_{ij}$ only at first order, and when non-Gaussian contributions cannot be neglected, ${\tilde{\mathcal{C}}}_{ij}$ receives quantum corrections, depending on the couplings in a non-trivial way. Inferring the Gaussian kernel ${\tilde{\mathcal{C}}}_{ij}$ from the knowledge of ${\mathcal{C}}_{ij}$ is a very challenging problem in field theory. In some approximation schemes however, relevant to extracting non-perturbative information about the behavior of the RG flow, this difficulty is not a limitation of our investigation. In the local potential approximation that we will consider in this paper for instance, we assume that ${\mathcal{C}}_{ij}^{-1}$ and ${\tilde{\mathcal{C}}}_{ij}^{-1}$ differ by a constant, ${\mathcal{C}}_{ij}^{-1}={\tilde{\mathcal{C}}}_{ij}^{-1}+k$; the constant k capturing all the quantum corrections. One would expect that an approximation works, essentially, in the region of small eigenvalues for ${\mathcal{C}}_{ij}^{-1}$, the IR regime in the field theoretical language. We hope that our methods can track the presence of a signal. We will return on this discussion in Section 3.1. A finer analysis would require more elaborate methods, beyond the scope of this paper. We are then able to construct an approximation of the RG flow, which is not autonomous due to the lack of dilatation invariance of the eigenvalue distribution for ${\mathcal{C}}_{ij}^{-1}$ (see Section 3). Finally, let us mention a remark about the field theoretical embedding. In Section 3, we show that even for the MP law, the number of relevant interaction becomes arbitrarily large in the first $66\%$ of the smallest eigenvalues. This introduces an unconventional difficulty in field theory, which can be alternatively viewed as a limitation of the field theory approximation. The breaking down of the field theory up to a certain scale is not a novelty. It is well known for instance that the Ising model behaves like an effective ${\varphi }^4$ field theory like (3) in the vicinity of the ferromagnetic transition. Thus, a failure of the field theoretical approximation may be alternatively viewed as a signal that a more elementary description is required. Then, it is interesting to remark that the field theory considered in (5) may be essentially deduced from the Ising-like model: where the $S_i=\pm 1$ are discrete Ising spins. Indeed, introducing N reals variables ${\varphi }_i$, and using the standard Gaussian trick to rewrite the quadratic term in $S_i$; Thus, summing over $\{S_i\}$ configurations generates an effective ${\sum }_{i}cosh({\varphi }_i)$; and expanding it in power of ${\varphi }_i$ reproduces the terms appearing in the local expansion in (5). The model described by (8) is reminiscent of the standard spin-glass models, as the Sherrington–Kirkpatrick model [54,55,56,57]. Its derivation, moreover, follows directly from the maximum entropy prescription with constraint (6) if we assume to work with discrete spins. We can alternatively see our model of field theory as coming from this binary model, derived from a principle of maximum entropy with fixed correlations [2,3,4].

In this section, we provide a mathematical definition of the field theoretical model that we consider, an extended discussion can be found in [20,23]. Following these references, we work in the eigenbasis of the matrix ${\mathcal{C}}_{ij}^{-1}$, more suitable for a coarse-graining approach. Note that, with our assumption, this eigenbasis is the same as the one for ${\tilde{\mathcal{C}}}_{ij}^{-1}$. In that way, the Gaussian (or kinetic) part of the classical action of $p[\varphi ]$ takes the form; where ${\lambda }_{\mu }$ denote the eigenvalues of ${\tilde{\mathcal{C}}}_{ij}^{-1}$, labeled with the discrete index $\mu$; and the fields $\{{\psi }_{\mu }\}$ are the eigen-components of the expansion of ${\varphi }_i$ along the normalized eigenbasis $u_i^{(\mu )}$: Let then $m^2$ be the smallest eigenvalue. We introduce the positive definite quantities $p_{\mu }^2:={\lambda }_{\mu }-m^2$. This way, the kinetic action takes the form of a standard kinetic action in field theory: In the continuum limit, for N sufficiently large, it is suitable to use the empirical eigenvalue distribution $\chi (\lambda )$ and to replace sums by integrals. This distribution is empirically inferred from ${\sum }_{\mu }\delta (\lambda -{\lambda }_{\mu })/N$, and the distribution $\rho (p_{\mu }^2)$ for the momenta $p_{\mu }^2$ follows. Moreover, with our assumptions on ${\tilde{\mathcal{C}}}_{ij}^{-1}$, this distribution can be directly deduced from the spectrum of ${\mathcal{C}}_{ij}^{-1}$. Finally, for purely random matrices with i.i.d entries, the MP theorem states that the empirical distribution has to converge toward an analytic form that can be used to do exact computations.

It is, however, difficult to deal with interactions in this formalism. A simplification, already considered by the authors of [23] is to work with momentum-dependent field $\psi (p)$ rather than with the field ${\psi }_{\mu }$ (labeled with the positive quantity $p_{\mu }^2$). This can be motivated by the observation that the model (5) is no rather fundamental than any other model able to reproduce effective correlations in some approximation scheme. In local approximation, that we consider in this paper, the two formulations are expected to be in the same universality class when we define locality in momentum space as:

This definition follows the standard one in classical and quantum field theory, and it is moreover consistent with the idea that locality is defined ‘‘at contact point” in the original representation (5).

Among the different incarnations of the Kadanoff–Wilson’s coarse-graining idea, the Wetterich–Morris (WM) framework has the advantage to be well suited to non-perturbative approximation methods [58,59]. Rather than Kadanoff–Wilson approach, which focuses on the effective classical action ${\mathcal{S}}_k$ for IR modes below the scale k, the WM formalism focuses on the effective averaged action ${\mathsf{\Gamma }}_k$; i.e., the effective action for integrated-out modes above the scale k. As recalled in the previous section, the fundamental ingredient to describe IR scales, when all degrees of freedom have been integrated out is the effective action $\mathsf{\Gamma }[M]$, which is defined as the Legendre transform of the free energy $W[j]$ (Equation (2)); the classical field $M=\{M_{\mu }\}$ being defined as: The starting point of the WM formalism is to modify the classical action $\mathcal{S}[\psi ]$ adding a scale dependents term $\Delta {\mathcal{S}}_k[\psi ]$, depending on a continuous index k running from $k=\mathsf{\Lambda }$, for some fundamental UV scale $\mathsf{\Lambda }$, to $k=0$. In such a way, we define a continuous family of models, described by a free energy $W_k[j]$ defined as: The regulator function $\Delta {\mathcal{S}}_k[\psi ]$ behaves like a mass, whose value depends on the momentum scale: The momenta scale $r_k(p_{\mu }^2)$ provides an operational description of the Kadanoff–Wilson’s coarse-graining procedure, and it is chosen, such that:

The two boundaries conditions ensure that we recover the effective descriptions, respectively, in the UV limit, where no fluctuations are integrated out, and in the deep IR where all the fluctuations are integrated out. In other words, we interpolate between the classical action $\mathcal{S}$ and the effective action $\mathsf{\Gamma }$. This can be achieved by introducing the effective averaged action ${\mathsf{\Gamma }}_k$ defined as: such that ${\mathsf{\Gamma }}_{k=0}\equiv \mathsf{\Gamma }$ and, from the conditions on $r_k$, ${\mathsf{\Gamma }}_{k=\mathsf{\Lambda }}\sim \mathcal{S}$. The meaning of ${\mathsf{\Gamma }}_k$ is illustrated in the Figure 3. Along the path from $k=\mathsf{\Lambda }$ to $k=0$, ${\mathsf{\Gamma }}_k$ goes through the theory space, and the different coupling changes. The dynamics of the couplings can be deduced considering a small variation $k\to k+dk$, and we can show that ${\mathsf{\Gamma }}_k$ obeys to the WM equation [58,59]: This equation is the one that we will use in this paper to investigate RG flow for datasets. The dot notation ${\dot{\mathsf{\Gamma }}}_k$ represents the partial derivation of ${\mathsf{\Gamma }}_k$ with respect to the scale k.