Mohamed Ouerfelli - Sum of Tensor Trace Invariants for Spin Glass Landscapes Optimisation

The presentation explores the optimization of spin glass landscapes using tensor trace invariants, emphasizing their application across fields like theoretical physics, artificial intelligence, and statistical physics. The speaker collaborates with a PhD student, M, and outlines a model involving a tensor with Gaussian noise and a signal. Key questions addressed include the detectability and recovery of signals, with a focus on computational efficiency. The discussion highlights the use of tensor invariants to enhance detection and recovery, comparing existing methods and exploring the potential of summing finite tensor invariants to approach theoretical limits in signal detection.

• The presentation focuses on optimizing spin glass landscapes using tensor trace tools.
• These tools have been previously discussed in seminars related to combinatorics and arithmetics in physics.
• The aim is to connect these mathematical models to fields like artificial intelligence and theoretical physics.

"So good afternoon everyone I'm very happy to be present with you today to present some of our work on optimization of spin glass landscapes using this tools which are transor Trace in."

• The speaker introduces the topic of optimization using tensor trace tools, highlighting its interdisciplinary applications.

• The model involves a random Gaussian tensor with a signal added, characterized by a signal-to-noise ratio (beta).
• The objective is to detect or compute the signal, which varies in difficulty based on the value of beta.
• Key questions include detection thresholds, recovery related to statistical physics, and efficient computation.

"We have a gan tonsor the noise in red here which is z, so every one of its component is a random Gan value and we add a signal to it."

• The model is defined as a Gaussian tensor with an added signal, forming the basis for the optimization problem.

• The model is connected to statistical physics through high-dimensional, non-convex landscapes with numerous minima.
• Such landscapes are common in glass systems, providing insights into phenomena like glassy transitions and aging.
• The model serves as a proxy to study complex systems in statistical physics.

"This is like this kind of f Scapes actually like happen in many important systems statical physics in particular the glass systems which is quite common actually in in general in life."

• The model's landscapes are analogous to those in glass systems, offering a simplified way to study complex physical phenomena.

• Advances in computing power allow for the analysis of data in tensor form, relevant to neural networks and multimodal data.
• The goal is to reduce computational costs and improve efficiency in applications like neural network layer completion.
• The work has implications for reducing energy consumption in AI systems.

"We also have like it's so it's not related to the to before but in some cases we can use actually we can use this uh this work that we did and we actually used it to do the completion of uh multiple like neural network layers because the layers are in the form of tonsor."

• The research has practical applications in optimizing neural network layers, highlighting its relevance to AI efficiency.

• The model reveals a gap between theoretical and practical detection thresholds, known as beta_stat and beta_comp.
• The gap indicates a difference between what is theoretically possible and what can be achieved with current algorithms.
• Understanding this gap is crucial for advancements in AI and computer science.

"We have another thr which we call bet comp for computability, which is the thres for which above which we there exist that have been proven with the theory theoretically proven algorithm that can work above the thresold with the algorithm."

• The gap between theoretical and practical thresholds is a significant challenge, with implications for computational theory.

• Tensor invariants are scalars obtained by contracting tensor copies, invariant under specific transformations.
• These invariants can be illustrated using graphs, aiding in the visualization and understanding of tensor properties.
• The study of tensor invariants provides tools for exploring and optimizing the model.

"Let's say that we have our Tor T which like um K AIS and we Define this transformation on this rotation group that for the Tor we multiply every one of its axis by this orthogonal Matrix."

• Tensor invariants offer a mathematical framework for analyzing and optimizing the model, crucial for understanding its properties.

• Tensor invariants are explored using Einstein notation and graph representation.
• The concept of contraction between tensor axes is introduced, where a sum over repeated indices implies a contraction.
• Tensor invariants are illustrated as trace graphs, simplifying the representation and calculation of tensor properties.

"In this way for this particular tring variant we have an illustration as a trace inine graph which is quite simple and very useful actually to use."

• The quote highlights the utility of trace graphs in simplifying tensor invariant calculations.

• Tensor invariants are compared to matrix operations, specifically the trace of matrix powers.
• The generalization from matrices to tensors allows for the computation of various properties, such as averages and variances.

"So in the case of Matrix when you can take the multiple the Tres of the power of the mates which gives you inv variance over another transformation and in the case of Tor so this is kind of its generalization."

• This quote explains how tensor invariants extend matrix operations to higher-dimensional data structures.

• The computation of tensor moments involves finding covering graphs and calculating their faces.
• The expectation of tensor invariants is derived from these covering graphs, contributing to understanding the tensor's statistical properties.

"So this is the expectation of this invariant when we have a g uh of course we can do it for expectation we can also do it for variance."

• The quote emphasizes the process of determining tensor expectations and variances using graph-based methods.

• Tensor networks are discussed as systems of interconnected tensors, with tensor invariants forming a subset.
• The invariants are more restrained, allowing for easier computation of properties and applications in signal processing.

"The T variance are like tonsor networks so tonsor networks are you take multiple copies of the tonsor you kind of like contract different indices but in this case it's more restrained classes."

• This quote clarifies the relationship between tensor networks and tensor invariants, highlighting their computational advantages.

• Tensor invariants can be used for signal detection by examining deviations in scalar distributions.
• Recovering a signal requires constructing a matrix from tensor invariants, achieved by cutting edges in the graph representation.

"If you want to detect signal it makes sense maybe like with the scalar to see the distribution of the scalar like it's the deviation from what we expected but if you want to recover the signal we need a vector."

• The quote underscores the distinction between detecting and recovering signals using tensor invariants.

• Graph-based methods decompose tensor invariants into contributions from noise and signal.
• Different scenarios are considered based on which component dominates, affecting signal detection and recovery.

"We can decompose a tracing variant as a sum of the Pure Noise Matrix so in variant Trace if it's a scalar plus other contributions that comes from the signal."

• This quote explains the decomposition of tensor invariants into noise and signal components.

• An algorithm is developed to compute tensor invariants, leveraging GPU parallelization for efficiency.
• The algorithm's speed and utility are emphasized, making it practical for large-scale computations.

"They can be computed and actually they are the good thing they are very easy to compute on a GPU because it's some some that you can parallelize with the GPU it can be extremely fast."

• The quote highlights the computational efficiency of the algorithm using GPU resources.

• Discusses the use of malonic invariant trace for detecting signals within a distribution.
• Highlights the importance of comparing computed values to expected distributions to identify anomalies.
• Introduces the use of Chip CH theorem to determine the probability of a spike's presence.

"If you know that the malonic invariant trace for Gan random gives a distribution like this and you compute your scaron, you find it very far from this distribution, then you can detect the signal."

• This quote explains that by comparing the computed value (scaron) to the expected distribution, one can detect anomalies or signals.

• Introduces the maximal single tast graph and generalized melon graph as tools for theoretical proofs.
• Discusses the limitations of current knowledge on these graphs' expectations beyond certain values.
• Highlights the role of these graphs in detecting signals and the efficiency of certain algorithms.

"We have a Conor about its expectation, but it's still not verified yet. We know that it's up 2K equal 12; this expectation is correct, but for above that, we don't know exactly."

• This quote indicates the uncertainty and current limitations in verifying the expectations of the maximal single tast graph.

• Describes the efficiency of different algorithms in detecting signals using various graphs.
• Mentions the effectiveness of the M graph and Ming graph in reaching computational thresholds.
• Discusses the potential for new algorithms to surpass existing methods.

"Using the algorithm two, it's the algorithm for the detection of the recovery. If you take the M graph, you can reach already like this computational F."

• This quote illustrates how the M graph, when used with algorithm two, effectively reaches a computational threshold for signal detection.

• Explains the versatility of the framework in incorporating and surpassing existing methods.
• Introduces T unfolding and homotopy methods as successful existing approaches.
• Highlights the potential of the framework to propose new, more effective methods.

"This Frameworks is able to not only incorporate existing methods but also propose new methods that work better."

• This quote emphasizes the framework's ability to integrate and improve upon current methodologies.

• Discusses the historical use of graphs and tensors in estimating symmetric G's highest EG value.
• Describes the generalization of the resolvent from matrix to tensor cases.
• Highlights the challenges in computing the resolvent and reaching statistical thresholds.

"Using this graph, you can have an estimation of the largest highest value of symmetric G."

• This quote shows the application of graphs in estimating significant values in symmetric G.

• Discusses the statistical beta and its implications in reaching computational thresholds.
• Highlights the challenges in computing certain functions and the potential for finite graph sums to approximate theoretical values.
• Explores the gap between theoretical and computable values and the ongoing efforts to bridge this gap.

"This resolvent is not computable, so we don't have it requires to compute in integral of over dimension of a space of Dimension n."

• This quote highlights the computational challenges associated with the resolvent and its implications for reaching statistical thresholds.

• Identifies the motivation to explore the gap between statistical computation and theoretical possibilities.
• Discusses various approaches to understanding and potentially overcoming this gap.
• Emphasizes the broader implications of finding solutions for computational science and applied mathematics.

"It's a statistical computation gap that exists, and this doesn't exist only in Tor PCA; it exists on also on all these methods."

• This quote underscores the widespread nature of the computational gap and the motivation to address it across various methods.

• The discussion focuses on optimizing graph summation by reducing redundancy and repetition in graph structures.
• The aim is to determine which graphs contribute significantly to results and how to weight them appropriately.
• The optimization process involves combinatorial studies to identify graphs that don't self-similarize, thereby simplifying calculations.

"We can reduce actually the numbers because there is some repetition and some redundancy in this graph."

• This quote highlights the effort to minimize redundancy in graph structures to streamline computational processes.

"When we sum these different graphs, what coefficient do you put to which one? Do you just put them all together with the same weight, or do you give a different weight?"

• The quote addresses the challenge of determining appropriate weights for each graph in the summation process to optimize outcomes.

• A new framework is introduced that utilizes smaller classes of tensor networks with advantageous properties for computing expectations and variances.
• This framework allows for generalizing results and achieving better results than existing methods, especially in cases with different dimensions.

"We have this new framework based on these tools on the sing variant that gives us this option of compute some expectation and variance and to have very easy theoretical proofs."

• The framework provides a simplified and efficient means of calculating expectations and variances, enhancing theoretical analysis.

"When we look at the two simplest T variants, they correspond to the two state-of-the-art methods that have been introduced from other backgrounds."

• The framework's simplest variants align with established methods, validating its effectiveness and potential for improved performance.

• The study examines the implications of summing a finite number of graphs versus an infinite number, focusing on accuracy and computational feasibility.
• Criteria are developed to reduce the number of graphs summed without compromising accuracy, aiming to approach statistical efficiency.

"What happens if we sum not an infinite number of them but just a finite one of them?"

• This quote explores the potential benefits of finite graph summation, balancing computational efficiency with accuracy.

"We have obtained some criteria showing that you can reduce the number of graphs and invariant that to sum without reducing the accuracy that you obtain of the sum."

• Criteria are established to optimize the balance between the number of graphs summed and the accuracy of results.

• The discussed tools and frameworks are applicable beyond high-energy physics, extending to neural networks and other fields.
• Collaboration with other disciplines highlights the versatility and utility of the developed methods.

"This tool also very powerful and can be very useful in many other applications."

• The methods' adaptability to various fields underscores their broad applicability and potential impact.

"We are actually using them in the collaboration of vasu with the S where they study more on the applicative fields."

• Active collaborations demonstrate the practical application and relevance of these methods in diverse research areas.

• The selection and weighting of graphs are crucial for optimizing results, with simple choices already yielding improvements over existing methods.
• More sophisticated selections are being explored to further enhance results, highlighting the importance of rigorous criteria.

"The idea of not just taking anyone but having some criteria to choose which one to use because indeed they are not all equivalent."

• The necessity of establishing criteria for graph selection to ensure optimal results is emphasized.

"We began looking at more sophisticated choices but already just with the very simple choice, we obtain the existing methods."

• Even basic selection criteria can outperform established methods, indicating potential for further advancements.

• Increasing the degree of invariants and graphs raises computational costs, motivating the need for efficient selection and weighting strategies.
• The balance between computational feasibility and accuracy remains a key challenge in implementing these methods.

"The more you increase the degree, the hardest it is, and that's also one of the motivations of what param did when he chose much smaller."

• The quote addresses the trade-off between computational complexity and accuracy, highlighting the need for efficient strategies.

"It's computable, but then you have this increase of costs, so it's also like motivate even more to try to make better choices."

• The challenge of managing computational costs while maintaining accuracy drives the development of better selection criteria.