The memory matrix

In order to answer this kind of questions, we can perform reconstructions by using not only the structure matrix [f.] but also an additional matrix [m ], of the same size, where we memorise  

The construction of the memory matrix is performed by taking as a starting point a totally blank matrix [m 0 = 0], and by applying the following operations  

The combination of a totally constrained algorithm with equations 3.7 of the memory matrix allows us to build, at each iteration, two very different matrices the structure matrix where the reconstruction appears, and the memory matrix where the parameters of the illegal values are gradually accumulated. 

If the distribution of these values were totally random, the memory matrix would tend to remain uniform, but in reality its behaviour is much more complex than that. At many points the illegal values do 

By indicating with VQ the negative vortices and with Vu the positive ones, the recognition of the vortices is performed, every T iterations, with the following criteria  

---

Figure 3.3 The errors produced by an iterative reconstruction algorithm have patterns which appear, at each iteration, completely random (A), but if successive patterns are memorised together, it is possible to observe regular structures appearing in the memory matrix (B).

Another important result is obtained by applying this method to pictures of many different kinds, because it has been noticed that the space distribution of the vortices is picture-dependent. The vortices pattern does not depend therefore on general characteristics of the algorithm, but on specific properties of the examined picture. It is as if a picture had a specific image in the memory space exactly as it has one in the real space. This brings us immediately to the following question Is it possible to use the information that appears in the memory matrix to improve the reconstruction in the structure matrix ... 

This result is interesting because it focuses our attention on the individual features of the memory matrix. If only vortices are memorized, it is obvious that the algorithm performs better with pictures that have a high potential number of vortices, but if other features could be memorized, it would become possible to reduce substantially the unknowns even with grey pictures. We have therefore the problem of discovering if other features exist which allow us to... 

At this point we are left with the problem of discovering yet more memory matrices, and here we have plenty of suggestions. It is plausible, for example, that a memory of boundaries, or more generally a memory of discontinuities, could be built, but we can leave these developments to the future. We have seen that the memory matrix method can indeed perform reconstructions from incomplete information, and therefore we already have what we were looking for a model that may help us understand the logic of embryonic development. 

The information that appears in the memory space cannot be transferred automatically to the structure space, and can be used only by employing specific conventions (the recognition of vortices in the memory matrix, for example, can be used only if a convention gives a meaning to the corresponding points of the structure matrix). This is another conclusion that leads to a universal principle, because it is necessarily valid for all systems. 

Integration of Eq. (11.4.1) then gives the relaxation matrix r in terms of the memory matrix K(f) as... 

Note that the time evolution of experimental observables zl (t) is governed by real dynamics which is determined according to Eq. 83 by the real propagator exp fit. The situation with the time evolution of the stochastic force Fn t) and the memory matrix Knk t) is much more complicated. According to Eq. 85, their evolution is governed by projected dynamics the propagator of which is given by exp iQLQt. ... 

Note that the time evolution of the generahzed stochastic Langevin force F (T f t) and the memory matrix are governed by so-called projected dy-... 

The analytical form of the memory matrix (f t — t) given in Eq. 101 is much less clear. The memory matrix (t t — t) contains all the (unknown) information about intermolecular interactions, i.e., all consequences of entanglement effects. The fluctuating part of the intermolecular force F t) acting on the n segment of the tagged chain can be expressed by matrix density fluctuations around this segment ... 

On this basis the memory matrix can be expressed as an integral over the time-dependent autocorrelation function of matrix density fluctuations for projected dynamics ... 

Taking all approximations together, the memory matrix can be expressed... 

The preaveraging approximation given in Eq. 107 is again used. The memory matrix then becomes a scalar relative to space rotations, and does not depend on the momentary configuration of the tagged chain ... 

---