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[MATH] instead of [MATH] . The advantage of [MATH] is that it represents the conditional probability of one observation [MATH] coming from the signal assuming a balanced mixture, and hence is bounded between zero and one. This greatly simplifies its visualisation and non-parametetric likelihood estimation. Taking Eq. 2... |
[MATH] by adding and subtracting [MATH] we have: [EQUATION] which can in turn can be expressed as: [EQUATION] hence proving that [MATH] |
is also a sufficient statistic and theoretically justifying its use for inference about [MATH] . The advantage of both [MATH] and |
[MATH] is they are one-dimensional and do not depend on the dimensionality of [MATH] hence allowing much more efficient non-parametric density estimation from simulated samples. Note that we have been only discussing sufficiency with respect to the mixture coefficients and not the additional distribution parameters |
[MATH] . In fact, if a subset of [MATH] parameters are also relevant for inference (e.g. they are nuisance parameters) then [MATH] and [MATH] are not sufficient statistics unless the [MATH] and [MATH] have very specific functional form that allows a similar factorisation. |
# Source: arxiv 1806.04747 # Title: Commutative Bezout domains of stable range 1.5 # Sections: all # Downloaded: 2026-03-03T05:19:28.330834+00:00 |
Commutative Bezout domains of stable range 1.5 Dedicated to the 70-th birthday of Professor V.V. Sergeichuk Abstract A ring [MATH] is said to be of stable range 1.5 if for each [MATH] and [MATH] satisfying [MATH] there exists [MATH] such that [MATH] . Let [MATH] be a commutative domain in which all finitely generated i... |
keywords: Commutative Bezout domain, Elementary divisor ring, Adequate ring, Stable range of a ring MSC: 13C05, 13G05, 16U10 Introduction and main results |
The problem of finding canonical forms of a matrix up to equivalency is classical. The rings over which matrices are equivalent to certain diagonal matrices have been studied extensively. In the present paper we investigate such question for some classes of commutative Bezout domains. |
The matrix [MATH] means a (possibly rectangular) matrix having [MATH] on main diagonal and zeros elsewhere (by the main diagonal we mean the one beginning at the upper left corner). We use the following notations of commutative rings: [MATH] denotes the greatest common divisor of elements [MATH] and [MATH] means that [... |
An associative (not necessary commutative) ring [MATH] is called an elementary divisor ring (introduced by I. Kaplansky in ) if every (not necessary square) matrix [MATH] over [MATH] admits a diagonal reduction, that is, there exist invertible matrices [MATH] and [MATH] over the ring [MATH] such that |
[EQUATION] in which each element [MATH] is a total divisor of [MATH] for [MATH] (i.e. [EQUATION] which is equivalent to [MATH] when [MATH] is commutative). The matrix [MATH] is called the Smith normal form and |
[MATH] are the invariant factors of the matrix [MATH] . Examples of such rings are the ring of integers [MATH] (see ), Euclidean rings and principal ideal rings (see |
). A ring [MATH] is a Bezout ring if each of its finitely generated ideals is principal. Each matrix from [MATH] and [MATH] over an elementary divisor ring [MATH] admits a diagonal reduction. This is equivalent to the condition that each finitely generated ideal in [MATH] is principal. Hence an elementary divisor ring ... |
showed that this condition can be replaced by the less restrictive hypothesis that [MATH] is adequate. A commutative Bezout domain [MATH] is adequate if for [MATH] with [MATH] , there exist [MATH] such that [MATH] , in which [MATH] and if [MATH] is a non-unit divisor of [MATH] , then [MATH] . Commutative principal idea... |
). The proof of the fact that an adequate ring is an elementary divisor ring (see , Theorem 3, p. 234] ) was based on , Theorem 1, p. 228] which says that if [MATH] has maximal rank over an adequate ring [MATH] , then there is a row [MATH] such that the [MATH] of the entries of [MATH] and the [MATH] of entries of the m... |
[MATH] such that [MATH] of entries of the matrix [MATH] and [MATH] of the elements of the first row of matrix [MATH] coincide. This result was generalized for matrices with rank greater then one by Petrychkovych in 15 , Lemma 3.1,p. 71] |
Deep studies of the theory of elementary divisor rings increasingly suggest that methods of pure ring theory are insufficient. Promising studies were based on the concept of stable range of rings, introduced by Bass |
as an important [MATH] -theory invariant. According to , Property [MATH] , p. 106] (see also the definition after Lemma 1 in ), the stable range of a ring [MATH] is the smallest positive integer [MATH] such that the following condition holds: |
[MATH] for each [MATH] satisfying [MATH] there exist [MATH] for which [EQUATION] If such [MATH] does not exist, then the stable range of [MATH] is infinity. |
The concept of the stable range of a ring turned out to be useful in the study of elementary divisor rings. In particular, Zabavsky 23 , Theorem 1, p. 666] proves that each elementary divisor ring has stable range [MATH] . His survey |
contains results on the problem when a commutative Bezout domain is an elementary divisor ring. We say that associative ring [MATH] has stable range 1.5 if for each [MATH] and [MATH] satisfying [MATH] there exists [MATH] with |
[EQUATION] This notion was introduced by the second author and studied in . Commutative principal ideal domains, adequate rings (see 14 , Prorositions 3.15 and 3.14] and , Proposition 4] ), rings of [MATH] matrices over rings listed before (see 18 , Theorem 5, p. 856] ) has stable range 1.5. |
Evidently each ring with stable range [MATH] has Bass stable range [MATH] . The converse is not always true. For instance, the subring [MATH] of the ring of formal power series [MATH] over the field of rational numbers [MATH] (see 10 , Example 1, p. 160] ) has stable range [MATH] but not [MATH] (see 17 , Example 1.1, p... |
Note that the notion of stable range 1.5 is closely related to the concept of almost stable range 1, introduced by McGovern A ring [MATH] has almost stable range 1 if each proper homomorphic image of [MATH] has stable range 1. In such rings if [MATH] , where [MATH] does not belong to the Jacobson radical [MATH] of [MAT... |
[EQUATION] In commutative rings of almost stable range 1 the condition [MATH] can be replaced to [MATH] (see , Proposition 4] ). Using this result and 14 , Theorem 3.7] we conclude that each commutative Bezout domain of stable range 1.5 is an elementary divisor ring. |
Several authors define and study rings of idempotent stable range , rings of unit stable range , rings of neat stable range , and rings of square stable range |
Our first result is a generalization of Helmer’s result , Theorem 1] Theorem 1 If [MATH] is a commutative Bezout domain, then the following conditions are equivalent: |
(i) [MATH] has stable range [MATH] (ii) for each [MATH] with [MATH] , there exists [MATH] such that [MATH] in which [MATH] coincides with the g.c.d. of entries of the matrix [MATH] |
The following example shows that the condition [MATH] in Theorem [MATH] is essential. Indeed, let [MATH] and [MATH] . Then [MATH] . The g.c.d. of entries of matrix [MATH] is equal to 1. But |
[EQUATION] for any [MATH] However the ring [MATH] has stable range [MATH] Matrices [MATH] and [MATH] satisfying equality ( ) are called transforming matrices of the matrix [MATH] . The set of all transforming matrices [MATH] and [MATH] are denoted by [MATH] and [MATH] , respectively. An elementary matrix is a matrix wh... |
Theorem 2 Let [MATH] be a commutative Bezout domain of stable range [MATH] . If [MATH] with [MATH] , then both of the sets [MATH] [MATH] contain an elementary matrices. |
Proofs If [MATH] are matrices such that [MATH] for some invertible matrices [MATH] and [MATH] , then we say [MATH] . We use the following result proved in 18 , Property 6, p. 50] |
Lemma 1 Let [MATH] be a commutative Bezout domain of stable range 1.5. Let [MATH] be a collection of relatively prime elements in [MATH] and [MATH] . Then there exist [MATH] such that |
(i) [MATH] (ii) [MATH] for each fixed [MATH] Proof of Theorem [MATH] . Let [MATH] has rank greater than 1. Without loss of generality, assume that [MATH] . A ring [MATH] is an elementary divisor ring. Therefore the equation ( ) holds for some invertible [MATH] and [MATH] . Since [MATH] , we have [MATH] in ( ). Consider |
[EQUATION] in which [MATH] [MATH] ) is the identity [MATH] matrix and [MATH] is an empty matrix. It is easy to check that [EQUATION] |
If [MATH] and [MATH] , then [MATH] and from [MATH] we obtain that [MATH] where [MATH] and [MATH] are the corresponding minors. There exist [MATH] by Lemma , such that |
[MATH] and [EQUATION] Moreover (see 18 , Property 3, p. 48] [EQUATION] for some [MATH] The equality [MATH] implies [MATH] , and so by ( ) we have |
[EQUATION] According to ( ), the greatest common divisor [MATH] of elements of the first row of the last matrix in ( ) is equal to |
[EQUATION] Using ( ) and reasoning as above, we obtain that [EQUATION] so [MATH] , in which [MATH] is the [MATH] of entries of [MATH] |
[MATH] . Let [MATH] , where [MATH] [MATH] , and [MATH] . There is [MATH] such that [MATH] , where [EQUATION] Hence [MATH] and so [MATH] has stable range 1.5. ∎ |
Proof of Theorem By Theorem , there exists [MATH] such that [MATH] , in which [MATH] is equal to the [MATH] of entries of [MATH] . Thus [MATH] is the first invariant factor of [MATH] (see ( )). Clearly |
[EQUATION] There exists an invertible [MATH] such that [MATH] Then [MATH] is also the [MATH] of entries of [MATH] . Thus, [MATH] for [MATH] and |
[EQUATION] Note that [MATH] and [MATH] are elementary matrices. Consider the submatrix [MATH] of [MATH] Using the same technique as above, we can find an elementary matrix [MATH] and an invertible matrix [MATH] such that |
[MATH] in which [MATH] is the second invariant factor of [MATH] Evidently [MATH] is elementary, [MATH] and [EQUATION] Continuing this process we obtain that there exist [MATH] and [MATH] such that [MATH] in which [MATH] is a product of elementary matrices. |
Taking [MATH] instead of [MATH] and applying the same reduction, we construct ”new” matrices [MATH] and [MATH] such that [MATH] is the Smith canonical matrix and [MATH] is a product of elementary matrices. Since [MATH] is symmetric, [MATH] , where [MATH] is a product of elementary matrices. |
Note that from Theorem does not imply that in ( ) both of [MATH] and [MATH] are elementary. Acknowledgement The authors are grateful for the referee’s valuable remarks and suggestions. The research was supported by the UAEU UPAR grant G00002160. |
# Source: arxiv 1806.04793 # Title: A Connectome Based Hexagonal Lattice Convolutional Network Model of the Drosophila Visual System # Sections: all # Downloaded: 2026-03-03T02:32:35.403330+00:00 |
A Connectome Based Hexagonal Lattice Convolutional Network Model of the Drosophila Visual System Abstract What can we learn from a connectome? We constructed a simplified model of the first two stages of the fly visual system, the lamina and medulla. The resulting hexagonal lattice convolutional network was trained usi... |
Introduction Universal function approximation results for artificial neural networks imply that many possible neural network architectures, with different connectivity matrices and different activation functions, can compute the same function. This suggests that it is difficult to predict the precise neural circuitry u... |
and Hassenstein-Reichardt , and how neural signals might be integrated . In contrast, since neural connectivity is generally sparse, knowledge of the connectivity of a neural circuit might constrain its possible computational function |
. Taken together, this suggests that it might be easier to predict function from structure, rather than the other way around. Recent reconstructions of the first two stages of the drosophila visual system |
enable us to test this hypothesis that neural circuit function can be predicted from neural circuit structure. We constructed a connectome-based artificial network model of the drosophila visual system and trained this model to track objects in videos of natural scenes. We found that a network model derived from the co... |
1.1 Prior work Existing work has focused on roughly correlating single cell responses in different brain brain regions to layers in a deep convolutional neural network |
, but not identify individual cell types and their connectivity. A different hexagonal lattice based model has also been proposed |
, however they do not model any circuitry beyond the photoreceptors, focusing only on the simulation of ommatidia. Specific to the Drosophila , mathematical models to fit the physiological motion responses have been suggested |
, however these simpler models were explicitly trained to learn the neural response. Other recent work tries to infer structure from function |
, without a connectome, and shows that a deep network can learn the neural responses of retinal ganglion cells when a convolutional network based encoding model is trained on natural images and white noise. They further show that recurrent lateral connections and feedforward inhibition help to reconstruct temporally ac... |
suggest that this is also the case for directional motion detection in the Drosophila Hexagonal lattice convolutional network model |
Based on publications of lamina and medulla connectomes, we modeled a connectome consisting of 43 neuron types. The neurons in these layers have a repeating columnar architecture with one column per ommatidium, each spanning [MATH] of visual angle, and they form a hexagonal lattice |
. Since the published connectomes only correspond to reconstructions of a few columns and not the entire hexagonal lattice, we repeat the locally described connectome in a spatially invariant manner, leading to a hexagonal lattice convolutional network. |
The network begins with the bundled rhabdomeres R1-R8. We simplified the neural superposition mapping by directly associating the inputs with their target column in the lamina and medulla, where they form inhibitory synapses |
. For the lamina, we simulate 10 neuron classes (L1-L5, C2-C3, T1, Lawf2, Am) that are known to contribute to motion processing. L1-L5, C2-C3 and T1 are synperiodic |
and therefore present in each column. Amacrine cells (Am) are multicolumnar. Lawf2 cells are likely presynaptic to most cells in the lamina, have wide receptive fields in the medulla and feedback to the lamina. As there are [MATH] Lawf2 neurons per optic lobe, and each column is innervated by [MATH] Lawf2 cells |
, we can distribute Lawf2 sparsely in our model (Figure , blacked out neurons are set to be constantly disabled). We had to omit Lawf1, Lai, Lat and glia cells due to a lack of dense reconstructions of their connectivity. |
For the medulla, we modeled the modular types identified in a dense 7 column (Figure (left)) reconstruction by Takemura et al . These include Mi1, Mi4, Mi9, T2, T2a, T3, Tm20, Tm1, Tm2, Tm4, Tm9, TmY5a, Dm8, Dm2 which can be mapped once per column |
. Additionally, we simulate T4a-T4d and T5a-T5d, which are ultraperiodic and connect with 4 distinct layers in the lobula plate . A recent update by Takemura et al |
looked at T4 connections beyond 7 columns and found a better connectome model (Figure 6a versus 6b previously), which also significantly improved our simulation results (Section ). For T5 however, we know that Tm1, Tm2 and Tm9 synapse onto T5, but the spatial configuration and delays are not yet known, since T5 resides... |
(Figure ), and we also simplified the ultraperiodic occurence of Tm3 to synperiodic. Many cells are transmedullar and project to the lobula or lobula plate |
. Transmedullar cells are the outputs of our simulated medulla and connect to a 3-stage fully connected decoder (Figure ), with 128, 32 and 4 neurons, which integrates visual information to solve our proxy-task (Section 4.1 ). |
Additionally, we used unpublished RNA sequence data to determine which neurons and synapses are inhibitory (such as Mi9) and excitatory, based on the neurotransmitters and receptors expressed. These constraints were also imposed during fine-tuning of the network weights (Section 3.2 ). |
The electron microscopic reconstructions contain detailed information about the morphologies of the neurons being modeled. However, we ignore these details, instead choosing to build a simplified network of linear-nonlinear point neurons with continuous non-negative activations and instantaneous synapses. Since neurons... |
We simulate a hexagonal lattice of [MATH] [MATH] point neurons with continuous activations, resulting in the typical [MATH] ommatidia and retinotopic columns found in a Drosophila fruit fly. Every neuron is defined as node with intrinsic properties: Name, sparsity along the [MATH] and [MATH] -axis, activation function ... |
2.1 Connectome-based network weights Connections between neurons in our model are defined as edge between source and target neuron. All weights are replicated spatially for all neurons of the same type, like in regular convolution filters. Depending on how two neuron types are connected laterally, arbitrary sparse filt... |
The complete model with all intercolumnar connection pattern and sparse filters can be found in the Supplemental Material . In Figure we present our complete network graph. Each edge starts at the receptive field, which is marked in the same color as the edge, and projects to its target neuron. Note that the hexagonal ... |
2.2 Physiology-based synaptic delays As we simulate our model as a convolutional neural network with recurrent components, the activity of every downstream neuron can only be computed when all inputs to a neuron are determined for step [MATH] . This means we needed to linearize the connections so that the network graph... |
Additional synaptic delays were introduced where physiological data suggested necessary delays: For T4, because of the temporal frequency optimum |
and the spatial offsets of Mi1 to Tm3 and only small temporal delay between them , we expect a delay at the negative input (Figure 6a ), which would form a more complex type of BL |
detector with additional properties not described by such simple models (Section 4.2 ). Our previous model based on older data suggested a configuration more akin to a HR |
model. For the T5 cells, due to the lack of a complete connectome, we assumed the same configuration of temporal and spatial offsets. |
2.3 Implementation In order to simulate our model efficiently, we used the tensor data structures and gradient based optimizers from the Caffe deep learning library |
combined with runtime code generating modules from LibDNN . We introduce a new layer, sparse repeated pattern recurrent neural network , which takes care of translating our connectome-based network defined in Python to runtime generated CUDA or OpenCL code. Caffe’s Python interface allows to conduct experiments rapidly... |
The newly introduced layer supports backpropagation through time (BPTT, ) during training, and a variable history length can be chosen. In our setup, we used a batchsize of 5 steps and a history of 10 steps. Images have to be resampled and rearranged for the hexagonal lattice (Figure (right)). To this end, we implement... |
. Note that we normalize the images and do not simulate any spherical projections that occur in real fly vision , since our focus is on the lamina and medulla circuitry. |
Gradient-based circuit optimization for object tracking In order to fine-tune the weights of our networks, we used a proxy-task that depends on the circuit’s ability to compute a moving object’s position and velocity. This approach is in contrast to explicitly training the network to learn an encoding model based on ph... |
, and thus requires only anatomical information, and no recordings of neural activity. 3.1 DAVIS dataset We used the DAVIS dataset |
in its 2016 version with 480p image resolution, which is used as a video object segmentation challenge. Due to its dense single-object annotation, it is easy to compute ground truth for the object centroid [MATH] and velocity [MATH] between consecutive frames. The network’s task is to predict these four parameters. The... |
3.2 Gradient based optimization of network parameters During circuit optimization, we used the Adam optimizer with weight decay to avoid large weights. The 164 decoder neurons were always initialized with Xavier |
weights without constraints. For our connectome-based network, six different training configurations were tested for functionality and robustness. |
Takemura 2017 (trained) Sign constrained weights initialized from the connectome. Takemura 2017 [MATH] 40% noise (trained) Initial weights perturbed with [MATH] uniformly sampled multiplicative noise. |
Takemura 2017 (untrained) Weights fixed to connectome, with only biases and decoder weights trained. Random (trained) Sign constrained weights initialized at random with MSRA |
Random [MATH] 40% noise (trained) Sign constrained weights using the previous MSRA initialization with [MATH] uniformly sampled multiplicative noise. |
Takemura 2013 (trained) Sign constrained weights using our older model for T4 inputs (Figure 6a ). For the random (MSRA ) initialized weights, the sign constraints were taken from the connectome initialized weights, since we only want to consider models which keep known inhibitory and excitatory properties. Bias values... |
optimization in step [MATH] . This keeps as many weights from being disabled as possible. Each configuration was trained for 30,000 iterations to minimize the squared error in predicting object location and velocity. |
Results 4.1 DAVIS object tracking Our networks are able to track objects with a mean [MATH] error as low as [MATH] in distance on the test set. The network which was constrained to only train the decoder and the bias values of the network was found to perform worst, with [MATH] error (Figure (left)). Predicted velociti... |
4.2 Predicted neural tuning properties We recorded all cell types both qualitatively as video clip (see Supplemental Material ) and quantitatively by checking for linear and nonlinear behavior. Most notably, we found that direction and orientation selective properties were recovered during training (Section 3.2 ). To c... |
A robust method based on vector addition was used to determine a noise-free selectivity index: [MATH] We found that training the Takemura 2017 initialized weight model with polarity constraints obtained from connectome data recovers and tunes T4 cells to well-known physiological responses |
. The neurons became highly direction selective in their preferred orientation (T4a: [MATH] , T4b: [MATH] , T4c: [MATH] and T4d: [MATH] , Figure 8e , angles are oriented as in Fisher’s measurements |
). They also show a two-lobed orientation selectivity with strong center-surround inhibition for bars orthogonal to the preferred direction (Figure 8f ). Previous models |
were also able to tune the cells, although less stable (Figures 8i 8j ). Networks initialized with random weights were unable to recover these properties (Figures 8g 8h ). The characteristic tuning properties were also absent with initial weights, however slight DS tuning is visible (Figures 8c 8d ). |
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