Why computers use matrices to do the differentiation of functions?
Computers use matrices for function differentiation because matrices offer a systematic, efficient way to handle vast amounts of data (like function inputs/outputs) and complex transformations, especially in multi-variable contexts (Machine Learning, Graphics). Matrices allow grouping partial derivatives (Jacobians) into single structures, simplifying notation, enabling powerful tools like automatic differentiation (AutoDiff) for neural networks (backpropagation), and leveraging highly optimized hardware for fast matrix operations (GPU).
Key Reasons Computers Use Matrices for Differentiation:
Data Representation: Matrices naturally organize large datasets, like inputs and gradients in neural networks, making complex calculations manageable.
Efficiency & Speed: Matrix operations (multiplication, addition) are highly parallelizable, allowing GPUs to perform thousands of calculations simultaneously, drastically speeding up deep learning training.
Generalization (Linear Algebra): Matrix calculus extends differentiation to multi-dimensional spaces (vectors/matrices), using tools like the Jacobian (all partial derivatives) and Hessian (second derivatives), which are fundamental for optimization.
Conciseness & Structure: A single matrix can represent many partial derivatives (like the Jacobian), making code cleaner and easier to understand than handling individual variables.
Automatic Differentiation (AutoDiff): Libraries like TensorFlow/PyTorch use matrix-based differentiation to automatically compute gradients, crucial for training models via backpropagation (calculating derivatives backward through the network).
Systematic Approach: Matrix-based methods provide a structured, algorithmic way to apply calculus rules (like the chain rule) to complex functions, which computers excel at.
To remember the chain rule, focus on the core idea: "Derivative of the outside, times the derivative of the inside".
Example: Neural Networks & Backpropagation
When training a neural network, you need the derivative of the loss (output) with respect to each weight (input).
This involves applying the chain rule across many layers.
Matrices (Jacobians) organize these derivatives, and backpropagation efficiently computes them by multiplying these matrices backward from the output to the input, updating weights systematically.
“Systems based on artificial intelligence (AI) are becoming ever more widely used for tasks from decoding genetic data to driving cars. But as the size of AI models and the extent of their use grows, both a performance ceiling and an energy wall are looming. With the performance of transistors in computer chips set to plateau, the computing power needed to support AI models will push today’s electronic hardware to its breaking point. Meanwhile, AI’s overall energy demand is soaring1, increasing carbon emissions and putting strain on local electricity grids around data centres. In September last year, technology firm Microsoft signed up for exclusive rights to the output of an entire US nuclear-power station to help to fuel its AI ambitions.
To enable a more sustainable future, the fundamental data-processing hardware requires a radical overhaul. Writing in Nature, Hua et al.2 and Ahmed et al.3 demonstrate complementary breakthroughs using silicon photonics — semiconductor chips that process light, rather than electricity — to increase computational performance, while decreasing energy consumption. Their electronic–photonic computing systems have key performance metrics that are comparable to, and in some cases surpass, purely electronic processors in real-world applications. This represents a significant leap towards finally capitalizing on the promise of photonic computing.
Photonic computing has been a topic of scientific research for decades, but real-world implementations have been held back by the lack of chip-scale, scalable optical analogues of the wildly successful silicon electronic integrated circuit. That has begun to change over the past decade, however, with the fabrication of high-performance photonic integrated circuits on the same silicon wafers as those used for microelectronics, largely motivated by the need for optical chip-to-chip interconnects.
Although several state-of-the-art demonstrations have shown the potential of integrated photonics to accelerate computation4–6, they have evaluated the performance of the photonic chips largely in isolation. But data in most real-world systems originate in the electronic domain, and so photonic computing requires close integration and co-design with electronics.
Indeed, the two technologies are complementary, rather than direct competitors. In particular, photonics performs linear operations, in which there is a simple proportional relationship between input and output data, more efficiently than electronics does. Electronics, meanwhile, excels at non-linear operations, in which the input and output data are related by more complex mathematical functions that do not preserve a proportional relationship between the two.
Photonic computing is particularly valuable for performing matrix multiplications, often referred to as multiply–accumulate (MAC) operations. These form the mathematical foundation of artificial neural networks and of many combinatorial optimization problems, which pop up frequently in areas such as resource allocation, network design, scheduling and supply-chain logistics. Solving such problems efficiently is a central priority for many modern computing systems, and is generally done by dedicated electronic computing accelerators, such as graphics processing units (GPUs) and tensor processing units (TPUs). These have highly parallel architectures that are better suited to calculating MAC operations than is the serial processing architecture of conventional central processing units (CPUs).
The latest work2,3 shows that silicon photonic computing can provide drastic improvements in key performance metrics while consuming less energy and being fully capable of running standard AI workloads. Hua et al.2, a Singapore-based team at the company Lightelligence, target combinatorial optimization problems, whereas Ahmed et al.3, at the firm Lightmatter in Mountain View, California, focus on running state-of-the-art AI workloads such as large language models.
Hua and colleagues2 apply their Photonic Arithmetic Computing Engine (PACE) to solve a class of combinatorial optimization challenges known as Ising problems, which have broad application in many real-world areas, including logistics and scheduling optimization. They directly compare its performance with that of a state-of-the-art NVIDIA A10 GPU for an Ising model that involves multiplying 64 × 64 matrices, and demonstrate a nearly 500-fold reduction in minimum latency — a key metric of computational speed — from 2,300 nanoseconds to just 5 nanoseconds. Furthermore, PACE has a latency scaling factor that is about 1,000 times smaller than that of TPUs, meaning that the performance gains in latency become even more pronounced as the matrix size is increased.
Ahmed and colleagues3, meanwhile, demonstrate a photonic processor that can execute standard state-of-the-art AI models — including the natural language processor BERT and convolutional neural network ResNet, used mainly for image recognition — with an accuracy close to that of standard electronic processors. The authors apply their photonic processor to an impressive breadth of real-world AI applications, including generating Shakespeare-like text, classifying film reviews as positive or negative and even playing the video game Pac-Man.
Screenshot of an old computer game called Pacman of a blue labyrinth with some blue ghost icons and yellow circles. Numbers with the score are indicated at the top in white.
Figure 1 | Winning quality. A game of Pac-Man was one of the applications on which the photonic processor of Ahmed et al.3 — one of two such recently developed systems2,3 — performed comparably to conventional electronic processors.Credit: ArcadeImages/Alamy
Despite these impressive breakthroughs, hurdles must still be overcome to realize the full potential of photonic computing as a commercial alternative to electronic accelerators. Much of the physical advantage of optical computing lies in its superior bandwidth and capacity for massive parallelism7. Both current demonstrations2,3 are limited by a clock speed — the number of operations that a processor can perform in one second — of the order of 1 gigahertz, whereas optical architectures and photonic devices can support speeds in excess of 100 gigahertz8 with minimal power dissipation. Furthermore, both demonstrations use monochromatic light in a single spatial waveguide mode. This leaves plenty of room for future improvements that could use many frequency and spatial modes in parallel. Finally, it remains to be seen whether these systems can maintain performance when they are scaled to the complex and dynamic workloads of commercial AI deployments.
Nevertheless, there are many reasons to expect that photonic computing accelerators will find their way into real-world systems in the near future. Crucially, the photonic and electronic chips used in both of these demonstrations were fabricated in standard, complementary metal-oxide-semiconductor (CMOS) foundries used to make microelectronic chips, and so this existing infrastructure can be immediately exploited to scale up manufacturing. Furthermore, both systems were fully integrated in a standard ‘motherboard’ interface, technically known as a peripheral component interconnect express interface, making them readily compatible with existing interfaces and protocols. Photonic computing has been in the making for decades, but these demonstrations might mean that we are finally about to harness the power of light to build more-powerful and energy-efficient computing systems.” [A]
A. Nature 640, 323-325 (2025) By Anthony Rizzo
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