MedQuest's Patel Rejects AI Black Box as Scaled Nonlinear Regression

Demystifying Deep Learning: A Conceptual Approach as a Function Approximator With MedQuest Imaging — Ai4

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Deep learning operates as transparent multivariable function approximation whose stacked nonlinear layers can safely mitigate clinical staffing deficits and medical errors once clinicians grasp its core mathematical mechanics.

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## Background Published on YouTube by Ai4 on August 28, 2025, this presentation features Dr. Aalpen Patel, Chief Clinical and Innovation Officer at MedQuest Imaging. The session is designed to demystify artificial intelligence and deep learning for healthcare professionals, translating complex mathematical concepts and black-box terminology into an intuitive framework of function approximation.

## TL;DR Dr. Aalpen Patel addresses the clinical urgency for AI adoption driven by acute workforce shortages and medical errors, while establishing clear distinctions between system transparency, interpretability, and explainability. He demonstrates that neural networks are continuous extensions of familiar function approximation techniques, tracing a direct conceptual path from Fourier transforms and multivariable clinical equations (such as the MELD score) to artificial neurons. Supported by the Universal Approximation Theorem and empirical simulations, Patel highlights how stacked layers and nonlinear activation functions like ReLU model intricate data geometries. Ultimately, he challenges the black-box myth, contending that deep learning is mathematically transparent and deterministic, even if massive parameter counts defy simple human mental parsing.

## Atmosphere & style The presentation carries an instructive, pragmatic, and reassuring tone, consciously stripping away generative AI hype in favor of foundational mathematical clarity. Speaking with the calm, pedagogical confidence of an experienced clinician, Dr. Patel uses familiar visual analogies and interactive simulations to dismantle technical intimidation. The resulting atmosphere feels empowering and grounded, transforming seemingly opaque machine learning concepts into a logical, accessible continuum of standard scientific problem-solving.

## Key takeaways - [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) Popular discourse often equates AI solely with generative tools like ChatGPT, but artificial neural networks remain the core mathematical building blocks across modern AI. - [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) Impending healthcare shortages—up to 86,000 physicians and 450,000 nurses—paired with roughly 250,000 annual deaths from preventable medical errors create an urgent clinical need for AI assistance. - [02:52](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=172s) AI evaluation requires distinct definitions: transparency covers underlying code and data, interpretability refers to human grasp of internal logic, and explainability means tracing specific outputs to inputs. - [04:49](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=289s) Function approximation is already standard across medicine and technology, from MP3 audio compression to MRI image reconstruction using Fourier transforms. - [07:21](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=441s) Statistical curve fitting and multivariable linear regression serve as everyday clinical tools, exemplified by log-transformed equations like the MELD score in liver disease. - [09:42](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=582s) An artificial neuron mirrors multivariable linear regression with the crucial addition of a nonlinear activation function to modulate signal flow. - [11:57](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=717s) The Universal Approximation Theorem proves feedforward networks can approximate any continuous function, with deeper multi-layer architectures achieving better accuracy with fewer parameters than single wide layers. - [14:46](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=886s) Nonlinear activation functions like ReLU or tanh are vital; without nonlinearity, cascading network layers collapse mathematically into simple linear regressors. - [16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) Deep learning is mathematically transparent because its architectures and algorithms are deterministic; the primary challenge lies in human interpretability across millions of parameters.

## Detailed walkthrough ### [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) Rethinking AI Beyond ChatGPT: Neural Networks as the Core Building Block

Aalpen Patel opens his presentation by thanking the organizers and Daniel Lackland for the invitation, signaling that his talk will pivot away from high-level hype to examine the technical fundamentals of deep learning. To gauge the audience's baseline familiarity, he surveys the room with a series of foundational machine learning concepts. He asks what comes to mind when hearing the term "AI"—whether attendees immediately think of ChatGPT, support vector machines (SVMs), random forests, or neural networks.

Seeing limited responses for classical machine learning methods, Patel observes that in popular discourse, "AI has been replaced by, is now synonymous with ChatGPT or the like." He emphasizes that despite the current generative AI craze, artificial neural networks remain the underlying mathematical and computational building blocks for virtually all modern breakthroughs. His stated goal for the session is to demystify these systems, strip away intimidating jargon, and replace black-box anxiety with conceptual comfort.

[01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) Healthcare Crisis: Workforce Shortages and Preventable Medical Errors

Establishing the clinical stakes, Patel highlights an impending crisis across the healthcare sector that directly impacts clinical capacity and diagnostic safety. Drawing on workforce projections, he outlines staggering labor deficits that are already underway: the United States faces an estimated shortage of up to 86,000 physicians by the year 2036, alongside projected deficits of 250,000 to 450,000 nurses. At the same time, an aging demographic demands an ever-increasing volume of diagnostic imaging, intensive treatments, and multifaceted clinical decision-making.

```mermaid flowchart TD accTitle: Healthcare Crisis Systemic Pressures accDescr: Diagram illustrating how workforce shortages and an aging population lead to diagnostic errors and the need for AI. subgraph Pressures["Systemic Pressures"] A["Workforce Shortages<br/>(86k Physicians, 450k Nurses)"] B["Aging Population &<br/>Increased Imaging Demand"] end subgraph Consequences["Clinical Impact"] C["Cognitive Overload &<br/>Complex Decision-Making"] D["Medical Errors<br/>(~250,000 Deaths/Year)"] end A --> C B --> C C --> D ```

Patel warns that this widening gap between clinical supply and patient demand creates the ideal conditions for medical errors to multiply. Referencing a landmark 2016 study from Johns Hopkins University, he underscores that approximately 250,000 Americans die annually from preventable medical errors—making it the third leading cause of death in the United States. He stresses that these numbers are unacceptable and that responsibly implemented AI tools will be essential to assist overburdened clinicians.

- **Physician Shortage:** Projected shortfall of up to 86,000 doctors by 2036. - **Nursing Shortage:** Estimated deficit ranging from 250,000 to 450,000 nurses. - **Johns Hopkins 2016 Study:** Medical errors account for roughly 250,000 patient deaths per year in the US.

[02:52](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=172s) Disentangling Transparency, Interpretability, Explainability, and the Black Box

Before diving into neural network mechanics, Patel insists on establishing precise definitions for terms that are frequently conflated across machine learning literature: transparency, interpretability, and explainability. He argues that vague language fuels unwarranted mystique around artificial intelligence.

```mermaid flowchart LR accTitle: AI Evaluation Dimensions accDescr: Three pillars defining how machine learning systems are evaluated: transparency, interpretability, and explainability. subgraph Dimensions["Core Definitions"] T["Transparency<br/>(Under the Hood: Data & Architecture)"] I["Interpretability<br/>(Human Understanding of Internal Logic)"] E["Explainability<br/>(Tracing Specific Outputs to Inputs)"] end ```

Patel unpacks each term through a practical lens: - **Transparency:** Concerns everything "under the hood"—the model's structural architecture, the curated training datasets, the optimization procedures, and the designated clinical use cases. - **Interpretability:** Pertains to human cognition—whether a human observer can inspect the internal mechanics of a model and comprehend the broad mathematical logic governing its decisions. - **Explainability:** Relates to post-hoc auditing of specific outputs—whether a clinician can trace a single recommendation or classification back to the exact input features that drove that outcome.

He clarifies that a system is only a "true black box" when proprietary vendor secrecy hides the architecture entirely, or when data dimensionality is so vast (millions of variables) that the resulting model complexity defies easy human parsing.

[04:49](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=289s) Function Approximation in Familiar Technologies: From MP3 to MRI

To ground deep learning in intuitive concepts, Patel frames neural networks as advanced tools for "function approximation"—the mathematical practice of fitting an estimated function to empirical data or mimicking a known continuous signal. He points out that society already relies on function approximation every day through digital signal processing.

MP3 audio compression, for instance, uses the Discrete Cosine Transform (DCT) to decompose sound waves into sinusoids, discarding high frequencies imperceptible to human ears to reduce file sizes. Similarly, JPEG compression applies lossy approximations to visual data, preserving what human vision registers while eliminating redundant details.

Moving to medicine, Patel illustrates how a Fourier series approximates a square wave by stacking successive harmonic frequencies; within roughly five sinusoidal terms, the synthesis closely resembles the target square wave. In Magnetic Resonance Imaging (MRI), scanners sample raw frequency data across k-space, using an inverse Fourier transform to approximate the spatial image. Low frequencies capture gross tissue contrast and structural outlines, while higher frequencies supply sharp edge details.

```mermaid flowchart LR accTitle: MRI Frequency Space Approximation accDescr: Low and high frequency components combining via inverse Fourier transform to create a diagnostic MRI. subgraph KSpace["k-space Sampling"] LF["Low Frequencies<br/>(Contrast & Fuzzy Shape)"] HF["High Frequencies<br/>(Fine Details & Edges)"] end subgraph Output["Image Reconstruction"] IFT["Inverse Fourier Transform"] MRI["Full Diagnostic MRI Image"] end LF --> IFT HF --> IFT IFT --> MRI ```

[07:21](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=441s) From Simple Linear Regression to Clinical Equations (MELD Score)

Patel transitions from digital signals to statistical modeling, reminding the audience that curve fitting is something clinicians and scientists have practiced since high school algebra through basic linear regression ($y = mx + b$). When expanded to multiple variables, a line in two dimensions becomes a flat hyperplane fitted through a high-dimensional cloud of data points, mathematically expressed as a weighted sum of inputs plus a bias constant:

$$\hat{y} = \sum_{i=1}^{n} w_i x_i + b$$

He demonstrates that input variables do not have to remain linear; they can be transformed using known functions. In hepatology, the Model for End-Stage Liver Disease (MELD) equation relies on log-transformed multivariable linear regression ($\ln(x)$) to evaluate liver dysfunction and stratify patient mortality risk. Patel notes that as an interventional radiologist performing Transjugular Intrahepatic Portosystemic Shunt (TIPS) procedures, calculating this approximate score is critical for determining whether a patient can safely tolerate the intervention.

- **Univariate Regression:** Standard slope-intercept form ($y = mx + b$). - **Multivariable Regression:** Linear combination of weights and features ($\sum w_i x_i + b$). - **Clinical Log-Transform Example:** The MELD score uses natural logarithms of lab values (bilirubin, creatinine, INR) to predict procedural outcomes in liver disease.

[09:42](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=582s) The Architectural Leap: Biological Neurons, Regressors, and Nonlinearity

Connecting statistical regression directly to artificial intelligence, Patel draws a structural parallel to a biological neuron. Dendrites serve as the input receivers, the soma (cell body) aggregates and modulates the incoming electrical signals, and the axon transmits the activated output downstream.

```mermaid flowchart LR accTitle: Artificial Neuron vs Linear Regressor accDescr: Flow diagram showing how input features are weighted, summed with bias, and transformed by an activation function. subgraph Inputs["Inputs & Weights"] X1["x₁ · w₁"] X2["x₂ · w₂"] X3["x₃ · w₃"] end subgraph Computation["Neuron Core"] SUM["Linear Combination<br/>∑(wᵢxᵢ) + b"] ACT["Nonlinear Activation Function<br/>f(z) [e.g., ReLU, tanh]"] end subgraph Result["Output"] OUT["Neuron Output (y)"] end X1 --> SUM X2 --> SUM X3 --> SUM SUM --> ACT ACT --> OUT ```

An artificial neuron performs almost the exact same calculation as a multivariable linear regressor, with one pivotal addition: the nonlinear activation function. The weighted inputs and bias are summed together, then passed through an activation function to prevent signal blow-up and inject nonlinearity.

When neurons are arranged in parallel and stacked into sequential layers, the network passes data from one hidden layer to the next. Patel describes deep learning as fundamentally a cascaded "function of a function of a function of a function." The deeper the network, the more intricate and abstract the representations it can approximate.

> "Each neuron is linear regressor plus the activation function, and then you add a whole bunch of neurons and that becomes your first layer... The deeper you go, that's the deep part of deep learning, the more complex patterns emerge."

[11:57](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=717s) The Universal Approximation Theorem and Experimental Simulations

Patel introduces the Universal Approximation Theorem as the core mathematical proof justifying deep learning. The theorem establishes that a feedforward network containing a single hidden layer with a sufficient number of non-linear neurons can approximate any continuous function on a compact domain to an arbitrary degree of accuracy.

To demonstrate how network width and depth affect approximation in practice, Patel shares empirical experiments where he attempted to fit a complex continuous wave signal split into training (blue) and testing (orange) points: - **1 Hidden Layer (30 Neurons):** Yielded a flat, completely inaccurate approximation that failed to capture the waveform. - **1 Hidden Layer (5,000 Neurons):** The network captured the gross contour of the function but remained coarse and missed sharp local variations. - **3 Hidden Layers (200, 500, 200 Neurons):** Achieved a substantially tighter fit with significantly fewer total parameters than the single 5,000-neuron layer. - **Deeper Multi-Layer Architectures:** Further improved edge resolution and fine details, though Patel observes that adding excess neurons eventually yields diminishing returns.

[14:46](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=886s) The Indispensable Role of Nonlinear Activation Functions

Reflecting on interactive demonstrations using the web-based TensorFlow Playground simulator, Patel explains what happens when training a neural network on complex geometric data, such as two interlocking, heart-shaped classification clusters.

If the network relies strictly on linear activation functions, it remains mathematically equivalent to a standard linear regressor regardless of how many layers are added; it collapses into a straight boundary line that cannot separate the shapes. However, once nonlinear activation functions such as the Rectified Linear Unit (ReLU) or hyperbolic tangent ($\tanh$) are introduced, the network easily wraps decision boundaries around complex, winding distributions. Patel stresses that nonlinearity is the absolute engine behind deep learning's power.

```mermaid flowchart TD accTitle: Impact of Activation Functions accDescr: Comparison of neural network capabilities with linear vs nonlinear activation functions. subgraph LinearAct["Linear Activation"] L1["Cascaded Linear Layers"] --> L2["Mathematical Collapse to Single Linear Equation"] L2 --> L3["Failure on Non-Separable / Complex Data"] end subgraph NonLinearAct["Nonlinear Activation (ReLU / tanh)"] NL1["Cascaded Nonlinear Transformations"] --> NL2["Arbitrary Decision Boundaries"] NL2 --> NL3["Universal Function Approximation Achieved"] end ```

[16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) Challenging the Black Box Myth: Deep Learning Is Mathematically Transparent

Patel concludes with what he acknowledges is a controversial stance: deep learning is not inherently opaque. When judged against the strict definition of transparency, neural networks are completely transparent. Practitioners know the exact matrix algebra, the specific layer topologies, the backpropagation loss algorithms, and the training datasets used to construct them.

The actual challenge lies in human interpretability and explainability. Because a deep network comprises millions of interconnected weights, humans cannot intuitively trace how an isolated parameter shifts an outcome the way they can inspect a single linear regression coefficient.

Nevertheless, Patel points out that modern diagnostic techniques, such as backpropagating gradients to generate saliency heatmaps, allow clinicians to see which visual features drove a specific model prediction. Deep learning is neither magic nor an impenetrable mystery—it is simply stacked, nonlinear regression operating at scale.

- **Mathematical Transparency:** Network architectures, loss functions, and optimization routines are fully known and deterministic. - **Interpretability Limits:** Massive parameter counts prevent simple mental modeling of high-dimensional interactions. - **Visual Explainability:** Attribution methods (e.g., gradient heatmaps) help trace specific outputs back to input features in clinical imaging.

## Notable quotes > [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) AI has been replaced by, is now synonymous with ChatGPT or the like. — Aalpen Patel

> AI tools are rapidly changing the way we test, diagnose, and treat patients. And while many clinicians have embraced them, they may not fully understand how they work. — Aalpen Patel

> In just 20 minutes, you’ll recognize the concept of deep learning is straightforward and intuitive, built on principles you already know. — Aalpen Patel

> Above all, we’ll show that deep learning is fundamentally a method of approximating relationships in data – a practice that’s longstanding and widely accepted in both medicine and scientific research. — Aalpen Patel

> [02:52](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=172s) true black box — Aalpen Patel

> [04:49](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=289s) function approximation — Aalpen Patel

> [09:42](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=582s) Each neuron is linear regressor plus the activation function, and then you add a whole bunch of neurons and that becomes your first layer... The deeper you go, that's the deep part of deep learning, the more complex patterns emerge. — Aalpen Patel

> [09:42](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=582s) function of a function of a function of a function. — Aalpen Patel

## Mentioned Aalpen Patel, MedQuest Imaging, Ai4, Daniel Lackland, ChatGPT, Johns Hopkins University, Discrete Cosine Transform (DCT), MP3, JPEG, Fourier transform, Magnetic Resonance Imaging (MRI), Model for End-Stage Liver Disease (MELD), Transjugular Intrahepatic Portosystemic Shunt (TIPS), Universal Approximation Theorem, Rectified Linear Unit (ReLU), TensorFlow Playground

## Video description Aalpen Patel, Chief Clinical and Innovation Officer, MedQuest Imaging

Abstracts

AI tools are rapidly changing the way we test, diagnose, and treat patients. And while many clinicians have embraced them, they may not fully understand how they work. AI is often seen as a mysterious black box, and the concept of deep learning – one of the cornerstones of AI – can be particularly intimidating.

If you can relate, this presentation is for you.

We’ll explain unfamiliar terminology and break down abstract mathematical concepts using visuals and simulations to help solidify your understanding. Above all, we’ll show that deep learning is fundamentally a method of approximating relationships in data – a practice that’s longstanding and widely accepted in both medicine and scientific research.

In just 20 minutes, you’ll recognize the concept of deep learning is straightforward and intuitive, built on principles you already know.

[Source](https://www.youtube.com/watch?v=0WS_WX2Zelc)

中文简报

MedQuest 的 Patel 驳斥 AI“黑箱论”：本质是规模化非线性回归 深度学习是一种透明的多变量函数逼近；临床医生一旦掌握其核心数学机制，其堆叠的非线性层便能安全缓解临床人手短缺与医疗差错。 Aalpen Patel 博士探讨了在严峻的人手短缺和医疗差错推动下应用 AI 的临床紧迫性，并明确区分了系统透明度、可解释性与可说明性。他论证了神经网络是常见函数逼近技术的延伸，梳理出一条从傅里叶变换和多变量临床公式（如 MELD 评分）通向人工神经元的直接概念脉络。在通用近似定理与实证模拟的支持下，Patel 指出堆叠层以及 ReLU 等非线性激活函数能够有效对复杂的数据几何形态建模。最终，他打破了“黑箱”迷思，主张深度学习在数学上是透明且确定性的，即便其庞大的参数规模超出了人类心智直接解析的能力。 面对预计到 2036 年美国高达 8.6 万名医生和 45 万名护士的劳动力缺口，以及每年约 25 万人死于可预防医疗差错的现状，医疗系统亟需 AI 工具来协助分担临床压力。 (1:43) 评估 AI 需严格区分三个维度：透明度关乎底层架构与训练数据，可解释性指人类对其内部运作逻辑的理解，而可说明性则是将特定预测追溯回具体输入特征。 (2:52) 人工神经元在数学上本质是叠加了非线性激活函数的多变量线性回归，堆叠多层神经元能够实现高维函数近似，类似临床上常见的 MELD 评分模型。 (9:42) ReLU 或 tanh 等非线性激活函数是深度学习的核心动力；若仅使用线性激活函数，无论堆叠多少网络层，在数学上都会塌缩为单一的简单线性回归。 (14:46) 深度学习在数学拓扑与算法层面完全透明，破除“黑盒”迷思的关键在于认识到其理解障碍主要源自数百万参数的高维交互，目前可通过梯度热图等方法实现输出归因。 (16:12)

Reading summary

Aalpen Patel argues that deep learning is not an inherently opaque mystery, but a mathematically transparent extension of multivariable regression and function approximation capable of supporting clinicians amid severe workforce shortages. Healthcare Shortages and Diagnostic Risk (1:43) Patel frames the clinical need for artificial intelligence around intensifying labor deficits and patient safety risks. Citing workforce projections, he outlines an expected shortfall of up to 86,000 physicians in the United States by 2036, alongside projected deficits of 250,000 to 450,000 nurses. These shortages intersect with an aging demographic that requires increasingly intensive treatment, complex clinical decision-making, and diagnostic imaging volume. According to Patel, the widening mismatch between clinical capacity and patient demand fosters conditions where diagnostic and management errors multiply. Referencing a 2016 Johns Hopkins University study, he points out that preventable medical errors account for roughly 250,000 deaths annually in the United States, making them the nation's third leading cause of death. He contends that responsibly integrated AI tools are needed to assist overburdened clinicians under these systemic pressures. Function Approximation from Signal Compression to MELD (4:49) To demystify neural networks, Patel presents deep learning as an advanced instrument for function approximation—the mathematical process of fitting an estimated function to empirical data or continuous signals. Everyday digital technologies rely on this principle: MP3 audio compression applies the Discrete Cosine Transform to discard imperceptible high frequencies, JPEG applies lossy compression to visual data, and Fourier series approximate square waves using stacked harmonics. Similarly, MRI scanners sample frequency data across k-space, utilizing an inverse Fourier transform to reconstruct diagnostic images by combining low-frequency tissue contrast with high-frequency structural edges. Clinicians already apply function approximation through statistical modeling and variable transformations. While standard multivariable linear regression fits hyperplanes using weighted inputs and bias constants, clinical scores frequently introduce nonlinear functions. In hepatology, the Model for End-Stage Liver Disease (MELD) score uses natural logarithms of lab values—bilirubin, creatinine, and INR—to quantify liver dysfunction and evaluate procedural mortality risk before interventions like Transjugular Intrahepatic Portosystemic Shunts (TIPS). Stacking Nonlinear Regressors for Complex Representations (9:42) Patel links statistical modeling to artificial intelligence by comparing an artificial neuron to a biological counterpart: dendrites receive inputs, the soma aggregates them, and the axon transmits the result. Computationally, an artificial neuron performs the exact operation of a multivariable linear regressor with one critical addition: a nonlinear activation function. Arranging neurons into sequential hidden layers makes deep learning a cascaded 'function of a function,' enabling the network to formulate increasingly abstract patterns. Nonlinearity is the vital mechanism that prevents deep architectures from collapsing. Demonstrations in the TensorFlow Playground simulator illustrate that networks using only linear activations reduce mathematically to a single linear equation, failing to separate non-linear data distributions. Once nonlinear functions like ReLU or tanh are applied, networks can wrap decision boundaries around complex shapes. Patel notes that while the Universal Approximation Theorem proves a single wide layer can approximate continuous functions, experimental simulations show that deeper, multilayer networks capture sharp details and local variations with significantly higher parameter efficiency. Distinguishing Mathematical Transparency from Cognitive Interpretability (2:52) Patel challenges the common portrayal of deep learning as an impenetrable 'black box' by separating the concepts of transparency, interpretability, and explainability. He defines transparency as full visibility into underlying architectures, training datasets, and optimization methods; interpretability as a human observer's ability to cognitively comprehend internal mechanics; and explainability as post-hoc auditing that traces a specific model recommendation back to its input features. Under these definitions, Patel contends that deep learning is mathematically transparent because the deterministic matrix algebra, layer arrangements, and loss-optimization algorithms are fully known. The true difficulty lies in human interpretability, as mental modeling cannot track interactions across millions of parameters. However, clinical explainability can still be achieved through diagnostic methods like backpropagating gradients into saliency heatmaps, which reveal the precise imaging features driving an algorithm's classification. Patel concludes that deep learning represents stacked, nonlinear regression operating at high dimensions rather than incomprehensible magic. While massive parameter counts limit direct mental interpretability, transparent mathematical foundations and visual explainability techniques allow these function approximators to be understood, audited, and safely utilized to alleviate clinical workloads.

精炼阅读摘要

Aalpen Patel 提出，医疗劳动力危机与高发的可预防医疗差错凸显了引入人工智能的紧迫性；他认为深度学习在本质上并非不可知晓的“黑箱”，而是层叠的非线性函数逼近器，其数学底层具有完全的透明度。 医疗劳动力赤字与临床差错压力 (1:43) Patel 强调，美国医疗系统正面临严峻的劳动力供给危机。行业预测显示，到 2036 年美国将面临多达 86,000 名医生的短缺，同时护士缺口预计在 250,000 至 450,000 人之间。与劳动力供给紧缺相对的是，人口老龄化带来了激增的影像检查需求与高强度的临床决策负荷。供需之间的鸿沟极易导致医疗失误增多。Patel 援引约翰斯·霍普金斯大学 2016 年的一项标志性研究指出，美国每年约有 250,000 名患者死于可预防的医疗差错，这已成为全美第三大死亡原因。他主张，以负责任的方式引入 AI 工具，已成为缓解临床人员超负荷运转、改善患者安全的必要手段。 函数逼近原理：从数字信号到临床回归 (4:49) 为了消除对深度学习的技术恐惧，Patel 将神经网络的基本逻辑解构为“函数逼近”，即拟合经验数据或重构连续信号的数学过程。现代技术早已广泛应用这一逻辑：MP3 音频压缩采用离散余弦变换（DCT）过滤人耳不敏感的高频信号，JPEG 图像格式亦采用有损压缩舍弃冗余视觉信息；在医学磁共振成像（MRI）中，设备在 k 空间采集频率数据，再通过傅里叶逆变换合成图像，其中低频信息构建组织对比与轮廓，高频信息填充边缘细节。在统计学中，这一过程体现为回归拟合。从单变量线性回归到高维超平面拟合，输入变量可以通过函数进行转换。例如肝病临床中的终末期肝病模型（MELD 评分），便运用自然对数转换的多变量线性回归评估死亡风险，指导经颈静脉肝内门体分流术（TIPS）等介入治疗决策。 网络架构与非线性激活函数的核心作用 (9:42) Patel 将人工神经元与生物神经元及多变量回归进行了映射对比：人工神经元的输入加权和偏置计算本质上就是多元线性回归，而加入的非线性激活函数（如 ReLU、tanh）则是实现深度学习突破的关键。如果仅采用线性激活函数，无论堆叠多少网络层，整个系统在数学上都会折叠为单一的线性方程，无法区分复杂的非线性几何分布；只有引入非线性转换，网络才能形成复杂的决策边界。通用近似定理从理论上证明了具有非线性激活单隐藏层的网络能够以任意精度逼近紧凑域上的连续函数。Patel 的拟合模拟进一步表明，仅靠增加单层神经元宽度（如 5,000 个神经元）效果粗糙，而采用三层深层网络（200、500、200 个神经元）能以更少的参数量实现更精确的拟合与细节分辨率，证明了多层函数级联映射高阶特征的有效性。 破除黑箱迷思：厘清透明度与可解释性 (2:52) Patel 明确区分了机器学习语境中常被混淆的三个概念：透明度涉及底层架构、训练数据集、优化程序及既定临床用途；可理解性指人类直觉认知模型决策内部数学逻辑的能力；可解释性则是指将具体输出结果追溯至特定输入特征的归因能力。他反驳了“深度学习本质上是黑箱”的观点，指出神经网络的基础矩阵代数、网络拓扑结构和反向传播算法完全确定且公开透明。造成黑箱印象的真正原因，在于数百万个连续权重交互所带来的高维复杂性超越了人类的心智推演极限，或是商业供应商的代码壁垒。通过利用梯度反向传播生成显著性热图等归因技术，临床医生已经能够追溯模型作出具体影像诊断的像素特征，证实深度学习并非神秘黑箱，而是规模化运行的多层非线性回归。 Patel 总结指出，深度学习是数学与计算原理完全可查的工程技术，而非神秘未知的智能黑箱。在医疗供给缺口扩大的背景下，厘清其作为非线性函数逼近器的本质，有助于临床人员卸下对算法不可知论的顾虑，更加科学、理性地利用多层网络处理复杂多维的诊疗数据。

Detailed notes — zh

## Background 该视频由 Ai4 频道于 2025 年 8 月 28 日发布在 YouTube 上。内容为 MedQuest Imaging 首席临床与创新官 Aalpen Patel 带来的主题演讲，旨在通过直观的视觉呈现与模拟实验，向临床医生和研究人员解构深度学习的底层逻辑，说明其本质是科学界熟知的“函数逼近器”。 ## TL;DR Patel 博士从医疗行业面临的人力危机与黑盒信任困境切入，通过 MP3 压缩、MRI 频域重构和临床 MELD 评分等熟悉案例，系统阐明了函数逼近的核心思想。他揭示了单个人工神经元本质上就是“多元线性回归与非线性激活函数”的结合，而深度网络则是这一结构的逐层复合，并受通用近似定理的理论支撑。通过拟合仿真实验，他直观展示了网络深度与宽度的权衡取舍以及非线性激活函数的决定性价值。最后，他澄清了深度学习在数学推导上其实完全透明，真正带来挑战的是超高维参数带来的可解释性难题。 ## Atmosphere & style 演讲语调沉稳亲切、循循善诱且充满临床共情，展现出极佳的教学感染力。主讲人通过生物神经元映射、生活化技术类比与动态曲线拟合模拟，将晦涩抽象的数学符号转化为极具画面感的故事线。整场分享节奏清晰、层层递进，有效消解了听众对前沿 AI 技术的畏难情绪，带来一种豁然开朗的认知掌控感。 ## Key takeaways - [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) 深度学习与神经网络是现代 AI 的底层基石，其本质并非神秘的魔法，而是建立在医疗与科学界熟知的经典统计学和数据拟合原理之上。 - [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) 医疗系统正面临严峻的医护人员短缺与高发医疗差错危机，AI 虽是弥补人力缺口的关键希望，但必须先解决由商业隐瞒或高维复杂度导致的“黑盒”信任问题。 - [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) AI 信任体系可严格拆解为三个维度：关注网络构造与训练数据的透明度（Transparency）、关注内部运作机制的可理解性（Interpretability），以及关注特定决策因果归因的可推导性（Explainability）。 - [05:06](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=306s) 函数逼近是数字时代与现代医学的核心基础，MP3 与 JPEG 的频域压缩以及 MRI（核磁共振）通过 K 空间采样和逆傅里叶变换重构图像均基于此原理。 - [07:21](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=441s) 从最简单的一维线性回归方程到高维数据拟合，临床常用的 MELD 肝病评分模型本质上就是加权回归与对数特征变换，证明医学决策长期以来一直依赖函数逼近思想。 - [09:54](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=594s) 人工神经元脱胎于生物神经元结构，其计算单元本质上由“多维线性回归器 + 非线性激活函数”构成。 - [09:54](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=594s) 激活函数带来的非线性转换是神经网络能够摆脱线性约束、拟合任意复杂高维曲面和非线性关系的核心枢纽。 - [11:27](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=687s) 深度学习的数学本质是“函数的逐层嵌套复合”，网络深度使模型能够逐层抽象并涌现出极其复杂的特征模式。 - [11:27](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=687s) 通用近似定理从理论上证明，只要包含足够数量的神经元并配合非线性激活函数，前馈神经网络就能在局部紧凑区间内以任意精度逼近任意连续函数。 - [12:48](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=768s) 拟合仿真实验表明，增加网络深度的效率远高于盲目增加单层宽度，多隐层架构能以更少的神经元总数更好地还原信号细节并抑制边缘效应。 - [14:46](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=886s) 若剥离非线性激活函数，无论堆叠多少层网络都会退化为普通的单层线性回归，彻底丧失对非线性数据边界的拟合与切分能力。 - [16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) 深度学习在数学原理、损失函数及反向传播机制上完全透明，其真正的黑盒挑战在于海量高维参数交互下的可解释性，目前可通过 Grad-CAM 等反向传播热力图对关键特征进行因果溯源。 ## Detailed walkthrough ### [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) 破除迷思：从 ChatGPT 到神经网络的本质 演讲开场，MedQuest Imaging 首席临床与创新官 Aalpen Patel 向大会组织方及 Daniel Lackland 表达了感谢，并迅速定下了本次分享的基调——不同于泛泛的高层次展望，他希望回归人工智能最底层的核心基石：深度学习与神经网络。为了了解现场听众的认知背景，Patel 博士向台下发起互动调查：“当我提到 AI 时，大家脑海里浮现的是什么？是 ChatGPT 吗？还是支持向量机（SVM）、随机森林（Random Forest），亦或是神经网络？”面对台下的反馈，他幽默而坦诚地指出，在当下的舆论环境中，AI 几乎已经被大众简单粗暴地等同于 ChatGPT 等生成式大语言模型，但实际上，神经网络才是支撑这一切复杂应用的底层积木。 Patel 博士明确了本次演讲的核心使命：用极其直观、剥离晦涩数学符号的方式，将深度学习层层拆解。他希望在短短二十分钟内，消除临床医生和研究人员对 AI 的畏难情绪，建立起对这一技术的掌控感与亲近感。深度学习绝非遥不可及的魔法，它的底层逻辑完全建立在医疗与科学界早已熟知并广泛应用的经典统计学与数据拟合原理之上。 --- ### [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) 医疗危机与黑盒困境：透明度、可解释性与可推导性 谈及在医疗领域引入 AI 的紧迫性，Patel 博士列举了一组令人警醒的数据：整个医疗系统正面临迫在眉睫的劳动力危机。预计到 2036 年，全美将面临高达 8.6 万名医生的缺口，而护士短缺数量据估计更在 25 万至 45 万人之间；与此相对的是人口老龄化加剧，对医学影像检查、日常护理及高复杂度临床决策的需求呈爆发式增长。超负荷运转直接推高了医疗差错风险——约翰斯·霍普金斯大学（Johns Hopkins）2016 年发表的一项里程碑式研究指出，美国每年约有 25 万人死于医疗差错，这已成为全美第三大死亡诱因。在这样的严峻背景下，AI 成为弥补人力鸿沟的关键希望，但前提是必须解决 AI 的“黑盒”信任危机。 ```mermaid flowchart TD accTitle: AI信任的三大支柱与黑盒定义 subgraph Pillars ["AI 信任的三大核心要素"] A["透明度 (Transparency)<br/>系统底层构造与训练细节"] B["可理解性 (Interpretability)<br/>人类能否看懂内部决策逻辑"] C["可推导性 (Explainability)<br/>特定决策能否追溯至输入特征"] end subgraph BlackBox ["黑盒形成的两种机制"] D["商业与信息黑盒<br/>厂商隐瞒数据与架构（三要素全缺）"] E["复杂度黑盒<br/>海量数据与多变量迫使模型极度复杂"] end Pillars -.->|任一条件缺失或受阻| BlackBox ``` 为了厘清业内常被混淆的概念，Patel 博士对 AI 的评估维度进行了严谨的拆解定义： - **透明度（Transparency）**：关注“引擎盖下是什么”，即模型是如何构建的、采用了何种网络架构、使用了哪些训练数据以及具体的使用场景。 - **可理解性（Interpretability）**：关注“人类如何理解其内部运作”，即人类专家能否直观领会模型内部的计算逻辑与决策规律。 - **可推导性（Explainability）**：关注“特定决策的因果溯源”，当模型给出某一特定诊断时，能否将其精确回溯到起决定性作用的原始输入特征上。 所谓的“黑盒”通常源自两种情况：一是供应商完全隐匿了算法和训练数据，导致上述三要素无一满足；二是面对海量高维变量时，为了提取有效特征而不得不采用极度复杂的深度模型，从而超出了人类直觉直接理解的范畴。 --- ### [05:06](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=306s) 什么是函数逼近？从 MP3、JPEG 到核磁共振（MRI） 为了破除对深度学习的恐惧，Patel 博士将核心概念引申至大家日常接触的经典技术——**函数逼近（Function Approximation）**。函数逼近的本质，就是寻找一个数学函数来最优拟合一组离散数据点，或极度逼真地模拟一个未知/已知函数。在日常数字生活中，这种思想无处不在： - **音频压缩（MP3）**：利用离散余弦变换（DCT），将声音分解为正弦波信号，剔除人耳无法感知的高频信息，从而在几乎不影响听感的前提下实现大幅压缩。 - **图像压缩（JPEG）**：采用有损压缩策略，舍弃人类视觉不敏感的冗余频域细节。 - **傅里叶级数（Fourier Series）逼近方波**：从单一的正弦波开始，随着叠加的高频分量增多，波形迅速趋近于标准的方波。通常只需叠加约 5 个频率分量，就能获得满足绝大多数工程需求的极佳逼近，完全无需无限计算。 ```mermaid flowchart LR accTitle: MRI成像中的逆傅里叶变换逼近过程 subgraph KSpace ["K 空间 / 频域采样"] LF["低频信号采样<br/>捕获整体轮廓与对比度"] HF["高频信号采样<br/>捕获边缘与精细解剖结构"] end subgraph Reconstruction ["逆傅里叶变换重构"] Combine["频域信息融合"] Image["清晰完整的临床 MRI 图像"] end LF --> Combine HF --> Combine Combine --> Image ``` 在医学影像领域，核磁共振（MRI）的成像正是函数逼近的典型范例。MRI 在采集过程中并不直接拍摄解剖切片，而是在 K 空间（频域空间）进行频率采样，随后通过逆傅里叶变换来重构逼近人体组织图像。如果仅采样低频信号，得到的图像将模糊不清，仅能辨别大致轮廓与明暗对比；如果仅采样高频信号，则只能看到锐利的组织边缘；只有将两者结合，才能逼近出高分辨率、具有完整临床诊断价值的 MRI 影像。 --- ### [07:21](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=441s) 从线性回归与 MELD 评分到多维拟合 从人们在高中时代就掌握的简单拟合出发，Patel 博士展示了函数逼近如何一步步向现实世界的复杂问题演进。当二维坐标系中散布着一组数据点时，人们习惯性地画出一条直线进行拟合，这就是最基础的线性回归，其数学表达式即为斜截式方程： $$y = mx + b$$ 当引入两个自变量时，二维的直线在三维空间中拓展为拟合点云的数据平面；当变量增加到数千乃至数百万个高维空间时，尽管人类大脑已无法将其可视化感知，但其代数形式依然只是权重与变量乘积的累加再加上偏置常数： $$y = \sum (w_i \cdot x_i) + b$$ ```mermaid flowchart LR accTitle: 线性拟合向非线性医疗评分模型的演进 A["一维线性拟合<br/>y = mx + b<br/>(二维直线)"] --> B["多维线性拟合<br/>y = ∑(w_i * x_i) + b<br/>(高维超平面)"] B --> C["广义对数变换回归<br/>y = ∑(w_i * ln(x_i)) + b<br/>(临床 MELD 评分)"] ``` 在临床医学中，这种数学形式早已被广泛应用。例如评估终末期肝病患者病情的 **MELD 评分（终末期肝病模型）**，其实质就是一种对数变换回归模型。在计算患者死亡风险时，该公式将胆红素、肌酐和凝血指标（INR）取自然对数后进行加权求和。Patel 博士结合自身临床工作指出，在进行经颈静脉肝内门体分流术（TIPS）这种肝脏血管介入手术时，MELD 评分是判断手术指征与评估预后的核心指标。这充分证明，临床医生每天都在依赖“加权回归与函数变换”进行重大决策。 --- ### [09:54](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=594s) 搭建桥梁：生物神经元、激活函数与多层神经网络 在铺垫了线性回归与特征变换后，Patel 博士揭示了从经典统计回归迈向现代深度学习的关键桥梁——**人工神经元（Artificial Neuron）**。人工神经元的设计灵感直接脱胎于生物神经元： - **树突（Dendrites）**：对应神经网络的**输入层（Input Layer）**，负责接收来自外界或其他神经元的信号输入。 - **细胞体（Cell Body / Soma）**：负责对所有输入的信号进行加权求和，并通过特定的生理阈值对累加信号进行非线性调制，防止电信号过大导致神经元“崩溃”。这个生理调节机制正是数学上的**激活函数（Activation Function）**。 - **轴突及末梢（Axon / Terminal Axon）**：负责将调制后的信号传递至下一级神经元，对应**输出层（Output）**。 ```mermaid flowchart LR accTitle: 生物神经元与人工神经元计算单元的映射关系 subgraph Biological ["生物神经元结构"] D["树突 (Dendrites)<br/>多渠道接收信号"] S["细胞体 (Soma)<br/>信号加权累加与生理抑制"] Ax["轴突 (Axon)<br/>向外输出动作电位"] D --> S --> Ax end subgraph Artificial ["人工神经元 (深度学习基本单元)"] X["输入特征与权重<br/>∑ (w_i * x_i) + b"] AF["激活函数 (Activation Function)<br/>注入非线性转换 (如 ReLU / Sigmoid)"] Out["节点输出<br/>传递至下一隐藏层"] X --> AF --> Out end Biological -.->|数学抽象与工程映射| Artificial ``` Patel 博士一针见血地指出，多元线性回归方程与单个人工神经元的数学形式几乎完全相同，二者唯一的根本区别就在于**激活函数**的引入。如果只有线性回归，无论叠加多少层网络，其整体依然只能表示线性关系；而正是激活函数带来的“非线性（Non-linearity）”，赋予了神经网络拟合任意复杂曲面与高维空间关系的强大能力。 单个神经元本质上就是“线性回归器 + 激活函数”。当成百上千个神经元并联排列时，便构成了深度神经网络的第一层；将第一层的输出作为下一组神经元的输入并逐层串联，便形成了包含多个隐藏层（Hidden Layers）的深度神经网络架构。深度学习的所有精妙能力，正是从这样一个个简单而经典的数学单元中层层涌现出来的。 ### [11:27](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=687s) 逐层复合与通用近似定理 在深度神经网络中，信息是按层级依次传递的：前一层的输出作为后一层的输入，第二层的输出又继续输入到第三层。从数学本质上看，深度学习就是“函数的函数的函数”（"function of a function of a function of a function"）。网络的层数越深——这正是“深度学习”（Deep Learning）中“深度”一词的由来——模型所能涌现和捕捉的特征模式就越复杂。其底层结构本质上是通过引入激活函数，将一系列线性回归器进行级联堆叠，而这种使用激活函数的机制甚至具有深厚的生物学神经元基础。 这一层级结构自然引出了著名的“通用近似定理”（Universal Approximation Theorem）。该定理指出：对于定义在有限紧凑域上的任意连续函数，只要在单隐层中提供足够数量的隐藏神经元，配合非线性激活函数，神经网络就能以任意精度逼近该函数。这是一个极其强大且具决定性的数学结论——尽管它并没有具体规定究竟需要多少神经元，也没有断言其他特定架构是否有效，但它从理论上保证了在特定领域内解的存在性。 - **逐层复合机制**：将多个线性回归器与激活函数级联，逐层传递非线性表征，使得深层网络能够表征极度复杂的模式。 - **通用近似定理的核心内涵**：只要神经元数量充足且存在非线性激活函数，单隐层网络理论上足以在局部有限区间内无限逼近任意连续函数。 ```mermaid flowchart LR accTitle: 深度神经网络的逐层函数复合机制 subgraph Layer1["输入与第一层"] X["输入数据 X"] --> F1["线性回归 + 激活: f1(X)"] end subgraph Layer2["第二层及深层网络"] F1 --> F2["线性回归 + 激活: f2(f1)"] F2 --> F3["线性回归 + 激活: f3(f2)"] end subgraph Output["最终预测"] F3 --> Y["逼近目标函数 Y"] end ``` ### [12:48](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=768s) 拟合仿真实验：增加宽度 vs 增加深度的取舍 为了直观展示这一理论，主讲人通过一个具体的已知连续信号拟合实验进行了演示。实验将蓝色数据点设为训练集，橙色数据点设为测试集，目标是让网络学会逼近该信号。在首次尝试中，模型仅使用包含 30 个神经元的单个隐层，结果拟合效果极差，根本无法反映信号规律；随后，主讲人将单层神经元数量大幅提升两个数量级至 5,000 个，此时网络虽然勉强勾勒出了信号的大致轮廓，但依然缺失了大量关键细节。 ```mermaid flowchart TD accTitle: 单层网络加宽与深层网络加深的逼近效果对比 subgraph Shallow["单隐层架构（增加宽度）"] S1["单层 30 神经元: 拟合严重失败"] --> S2["单层 5,000 神经元: 仅具粗略轮廓，缺失细节"] end subgraph Deep["多隐层架构（增加深度）"] D1["三层 (200-500-200): 神经元总数更少，但细节明显丰富"] --> D2["增加层数与节点: 边缘效应显著改善，逐步趋向边际效应递减"] end ``` 随后，主讲人调整了网络架构，引入多层隐层设计。实验证明，多层网络所需的神经元总数远少于 5,000 个，却能取得更好的拟合效果。例如，采用包含三层隐层（神经元数量分别为 200、500、200）的架构后，拟合曲线已经高度还原了原始信号的真实形态；在此基础上继续增加层数与神经元数量，模型对“边缘效应”（Edge Effects）的拟合精度得到显著改善，边界区域的刻画更加细腻。然而，网络规模扩大到一定程度后，继续堆叠参数会遭遇边际效益递减（Diminishing Returns）。 - **单层 30 神经元**：容量过小，拟合完全失效。 - **单层 5,000 神经元**：能够拟合整体趋势，但缺乏高频细节，单靠加宽单层效率低下。 - **三层架构（200-500-200 神经元）**：以更少的参数总量还原出更丰富的形态与细节。 - **更深层架构**：边缘和边界逼近能力更强，随后性能提升逐渐进入平台期。 ### [14:46](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=886s) 非线性激活函数的决定性作用与直观模拟 谈到逼近能力的本质，主讲人特别强调了非线性激活函数的决定性地位，并推荐使用 TensorFlow Playground 等在线深度学习仿真工具来建立直观认识。以二维平面上分离形似心形的数据集分类任务为例：如果网络采用线性激活函数，无论堆叠多少层，模型在达到某个极其有限的效果后就会停滞不前，根本无法划分复杂的非线性边界。 只有引入高级非线性激活函数——例如 ReLU（Rectified Linear Unit，整流线性单元）或双曲正切函数（tanh）——网络的非线性拟合能力才会被真正释放，从而精准地逼近复杂的几何分布。主讲人着重指出，通用近似定理成立的关键前提就是“激活函数必须具备非线性”；一旦剥离了非线性，多层网络就会退化为普通的单层线性回归，彻底丧失深度学习强大的表达能力。 - **线性激活的局限**：多层线性变换复合后依然是线性变换，无法切分非线性数据边界。 - **常用非线性激活函数**：ReLU、tanh 等能够赋予网络弯曲、折叠特征空间的能力。 - **交互式工具建议**：借助 TensorFlow 的可视化模拟器调整层数与激活函数，可以直观观察分类边界的动态演变。 ### [16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) 深度学习是黑盒吗？透明度与可解释性的辨析与总结 在演讲尾声，主讲人提出了一个颠覆常规认知的观点：学术界和工业界常将深度学习称为无法看透的“黑盒”，但从数学本质上讲，深度学习是“透明”（Transparent）的。我们完全掌握其背后的微积分原理、损失函数定义、反向传播权重更新机制以及每一层的网络架构与内部运作方式。 真正引发困扰的不是“透明性”，而是模型的“可解释性”（Explainability / Interpretability）。当模型包含数百万甚至数十亿个参数时，其复杂度超出了人类大脑直接理解决策因果的能力。相比于单一线性回归方程中清晰明了的自变量回归系数，深层复合网络的归因变得极具挑战性。尽管如此，业内已经发展出了通过反向传播生成热力图（如 Grad-CAM 等归因技术）等手段，用于探测驱动决策的关键输入特征。主讲人总结道，从最基础的线性回归，到单个神经元，再到深层神经网络，它们在数学逻辑上是一脉相承的函数逼近器，深度学习绝非不可捉摸的魔法。 - **透明性 vs 可解释性**：深度学习在数学推导与训练机制上完全透明，但在高维复杂参数交互层面难以直观解释。 - **归因与窥探手段**：可利用反向传播生成注意力热力图（Heat maps），定位影响模型判决的核心输入区域。 - **演讲核心脉络**：线性回归方程 $\rightarrow$ 结合激活函数的单神经元 $\rightarrow$ 多层深度神经网络，全流程均构建在严谨的函数拟合基石之上。 ## Notable quotes > [00:04](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=4s) 当我提到 AI 时，大家脑海里浮现的是什么？是 ChatGPT 吗？还是支持向量机（SVM）、随机森林（Random Forest），亦或是神经网络？ — Aalpen Patel > [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) 透明度（Transparency）：关注‘引擎盖下是什么’，即模型是如何构建的、采用了何种网络架构、使用了哪些训练数据以及具体的使用场景。 — Aalpen Patel > [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) 可理解性（Interpretability）：关注‘人类如何理解其内部运作’，即人类专家能否直观领会模型内部的计算逻辑与决策规律。 — Aalpen Patel > [01:43](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=103s) 可推导性（Explainability）：关注‘特定决策的因果溯源’，当模型给出某一特定诊断时，能否将其精确回溯到起决定性作用的原始输入特征上。 — Aalpen Patel > [05:06](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=306s) 函数逼近的本质，就是寻找一个数学函数来最优拟合一组离散数据点，或极度逼真地模拟一个未知/已知函数。 — Aalpen Patel > [09:54](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=594s) 多元线性回归方程与单个人工神经元的数学形式几乎完全相同，二者唯一的根本区别就在于激活函数的引入。 — Aalpen Patel > [11:27](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=687s) 深度学习就是‘函数的函数的函数’（function of a function of a function of a function）。 — Aalpen Patel > [11:27](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=687s) 对于定义在有限紧凑域上的任意连续函数，只要在单隐层中提供足够数量的隐藏神经元，配合非线性激活函数，神经网络就能以任意精度逼近该函数。 — Aalpen Patel > [14:46](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=886s) 通用近似定理成立的关键前提就是‘激活函数必须具备非线性’；一旦剥离了非线性，多层网络就会退化为普通的单层线性回归，彻底丧失深度学习强大的表达能力。 — Aalpen Patel > [16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) 从数学本质上讲，深度学习是‘透明’（Transparent）的。我们完全掌握其背后的微积分原理、损失函数定义、反向传播权重更新机制以及每一层的网络架构与内部运作方式。 — Aalpen Patel > [16:12](https://www.youtube.com/watch?v=0WS_WX2Zelc&t=972s) 真正引发困扰的不是‘透明性’，而是模型的‘可解释性’（Explainability / Interpretability）。当模型包含数百万甚至数十亿个参数时，其复杂度超出了人类大脑直接理解决策因果的能力。 — Aalpen Patel ## Mentioned Aalpen Patel, Daniel Lackland, MedQuest Imaging, Ai4, ChatGPT, 支持向量机 (SVM), 随机森林 (Random Forest), 约翰斯·霍普金斯大学 (Johns Hopkins University), MP3 / 离散余弦变换 (DCT), JPEG, 傅里叶级数 / 逆傅里叶变换 (Fourier Series), 核磁共振 (MRI / K-Space), MELD 评分 (终末期肝病模型), 经颈静脉肝内门体分流术 (TIPS), 通用近似定理 (Universal Approximation Theorem), TensorFlow Playground, ReLU, Grad-CAM ## Video description Aalpen Patel, Chief Clinical and Innovation Officer, MedQuest Imaging Abstracts AI tools are rapidly changing the way we test, diagnose, and treat patients. And while many clinicians have embraced them, they may not fully understand how they work. AI is often seen as a mysterious black box, and the concept of deep learning – one of the cornerstones of AI – can be particularly intimidating. If you can relate, this presentation is for you. We’ll explain unfamiliar terminology and break down abstract mathematical concepts using visuals and simulations to help solidify your understanding. Above all, we’ll show that deep learning is fundamentally a method of approximating relationships in data – a practice that’s longstanding and widely accepted in both medicine and scientific research. In just 20 minutes, you’ll recognize the concept of deep learning is straightforward and intuitive, built on principles you already know. [Source](https://www.youtube.com/watch?v=0WS_WX2Zelc)

Source quotes

[0:04] Aalpen Patel: 当我提到 AI 时，大家脑海里浮现的是什么？是 ChatGPT 吗？还是支持向量机（SVM）、随机森林（Random Forest），亦或是神经网络？ [1:43] Aalpen Patel: 透明度（Transparency）：关注‘引擎盖下是什么’，即模型是如何构建的、采用了何种网络架构、使用了哪些训练数据以及具体的使用场景。 [1:43] Aalpen Patel: 可理解性（Interpretability）：关注‘人类如何理解其内部运作’，即人类专家能否直观领会模型内部的计算逻辑与决策规律。 [1:43] Aalpen Patel: 可推导性（Explainability）：关注‘特定决策的因果溯源’，当模型给出某一特定诊断时，能否将其精确回溯到起决定性作用的原始输入特征上。 [5:06] Aalpen Patel: 函数逼近的本质，就是寻找一个数学函数来最优拟合一组离散数据点，或极度逼真地模拟一个未知/已知函数。 [9:54] Aalpen Patel: 多元线性回归方程与单个人工神经元的数学形式几乎完全相同，二者唯一的根本区别就在于激活函数的引入。 [11:27] Aalpen Patel: 深度学习就是‘函数的函数的函数’（function of a function of a function of a function）。 [11:27] Aalpen Patel: 对于定义在有限紧凑域上的任意连续函数，只要在单隐层中提供足够数量的隐藏神经元，配合非线性激活函数，神经网络就能以任意精度逼近该函数。 [14:46] Aalpen Patel: 通用近似定理成立的关键前提就是‘激活函数必须具备非线性’；一旦剥离了非线性，多层网络就会退化为普通的单层线性回归，彻底丧失深度学习强大的表达能力。 [16:12] Aalpen Patel: 从数学本质上讲，深度学习是‘透明’（Transparent）的。我们完全掌握其背后的微积分原理、损失函数定义、反向传播权重更新机制以及每一层的网络架构与内部运作方式。 [16:12] Aalpen Patel: 真正引发困扰的不是‘透明性’，而是模型的‘可解释性’（Explainability / Interpretability）。当模型包含数百万甚至数十亿个参数时，其复杂度超出了人类大脑直接理解决策因果的能力。

Source quotes

[0:04] Aalpen Patel: AI has been replaced by, is now synonymous with ChatGPT or the like. Aalpen Patel: AI tools are rapidly changing the way we test, diagnose, and treat patients. And while many clinicians have embraced them, they may not fully understand how they work. Aalpen Patel: In just 20 minutes, you’ll recognize the concept of deep learning is straightforward and intuitive, built on principles you already know. Aalpen Patel: Above all, we’ll show that deep learning is fundamentally a method of approximating relationships in data – a practice that’s longstanding and widely accepted in both medicine and scientific research. [2:52] Aalpen Patel: true black box [4:49] Aalpen Patel: function approximation [9:42] Aalpen Patel: Each neuron is linear regressor plus the activation function, and then you add a whole bunch of neurons and that becomes your first layer... The deeper you go, that's the deep part of deep learning, the more complex patterns emerge. [9:42] Aalpen Patel: function of a function of a function of a function.

Original transcript

[0:04] And I'd like to thank uh the organizers [0:06] and Daniel Lackland for inviting me. Uh [0:08] it's going to be a good talk. So my bend [0:11] is going to be a little different than [0:12] what we talked about before. I'm going [0:14] to focus on the fundamentals um of deep [0:17] learning and neural networks tend to be [0:20] building block for many of the things we [0:21] were talking about. So just to get a get [0:23] an understanding of the audience, how [0:25] many of you know uh when I when I say um [0:28] deep learning or when I say let's let's [0:31] back up even a little bit more. When I [0:33] say AI, what does that mean to you? Does [0:34] that mean to chat GPT? [0:39] Anyone? No. Does that mean support [0:42] vector machines? [0:46] Does that mean a random forest? [0:52] Okay. [0:54] Um, what about uh what about neural [0:57] networks? [1:00] Okay. [1:01] So, I'm glad um because when I talked to [1:05] a lot of folks uh the it seems that AI [1:09] has been replaced by as is now [1:11] synonymous with chat GPT or or the like. [1:14] So, um but neural networks tend to be [1:17] the building blocks of many of the [1:18] things we do. So, that's why I'm [1:19] focusing on that. uh and um and I hope [1:23] by the end of the day end of the uh [1:25] lecture um talk about um uh and give you [1:29] more comfort with with what we uh what [1:31] it is. So I hope to simplify and break [1:33] down the complex concepts demestify and [1:36] then elevate or increase your comfort [1:38] level and deeper of uh deep learning. [1:43] So in healthcare we have a impeding [1:44] healthcare crisis. How many of you are [1:46] in healthcare? [1:48] So you know you know what I'm talking [1:49] about. Um and um [1:53] so that healthcare uh crisis translates [1:55] into workforce challenges. There is a [1:59] significant amount of workforce [2:00] challenges. We have up to 86,000 [2:04] physician shortage coming by year 2036 [2:07] and some say the numbers are even higher [2:09] by 250 or 450,000 nursing shortages and [2:13] that's only going to increase over time. [2:15] Our population is aging, requiring more [2:17] imaging, more care, and more complex [2:21] decision-m. So all of this is conspiring [2:24] uh to potentially increase in the more [2:26] errors we already make. There's a paper [2:28] in 2016 out of Hopkins that said we [2:31] probably the uh about quarter of a [2:35] million people die of medical errors per [2:37] year. That's the third largest cause of [2:39] death in America. That's a staggering [2:41] number and it's an unacceptable number [2:44] and it can increase hopefully not but [2:47] but it's all the conditions are right to [2:49] increase if we don't do something about [2:51] this transparency interpretability and [2:54] explanability [2:56] and the reason I'm I'm defining this is [2:58] because what my last statement would be [3:01] uh it's critical on the definitions um [3:04] when you when you talk to folks or even [3:06] in the literature a lot of this is [3:07] conflated and that's why distinct [3:09] definitions are going to be important. [3:11] So what is transparency? The way I look [3:13] at transparency is how we built it, [3:16] what's what's the architecture, what the [3:18] data were used, how we trained it, uh [3:21] what's the use cases. So everything [3:23] that's under the hood. [3:26] Interpretability is um how it works, how [3:31] how can humans understand what's inside [3:33] and how it's making the decisions, [3:34] right? So it's about humans, humans [3:36] interpreting what's inside. And the [3:38] third is explanability and that's about [3:41] if a particular decision is made can I [3:43] trace it back to the inputs that guided [3:44] that guided that decision. So it's about [3:47] explaining a particular decision. So [3:50] what's under the hood? Do we know? Yes. [3:52] Can we understand it? Maybe. Uh and can [3:56] we explain a particular decision? [4:02] I'm also going to simplify the black [4:04] box. So what exactly is a black box? If [4:06] feed doesn't meet any of those three [4:08] conditions we talked about, it's a it [4:09] can be a black box. If it's not [4:12] transparent, explainable, or um um or uh [4:18] interpretable. So, it's a true black [4:20] box. Well, if a vendor doesn't tell you [4:22] what's in it, it's a true black box [4:24] because you have none of those three [4:25] conditions are met. [4:27] Then you have the complexity of the [4:28] data. If you have millions of variables [4:31] and a large volume of data, in order to [4:34] actually get anything meaningful out of [4:36] that, you're going to have to have a [4:37] complex model. And that complexity [4:39] sometimes uh uh leads to not being able [4:42] to explain or to understand it clearly. [4:49] So I'm going to try to evolve this from [4:51] what we already know, what we use what [4:53] knowingly or unknowingly in in our daily [4:55] lives or in our professional lives uh [4:58] into something that is uh that is neural [5:01] network and that is um uh and how [5:03] they're linked. [5:06] So the concept of functional [5:07] approximation that's what I'm that's [5:09] what I'm talking about. And uh function [5:12] approximation is a process of estimating [5:13] a function that best fits the data [5:15] points or can mimic can closely mimic a [5:18] function a known function. [5:22] So when we look at MP3s, MP3 uses [5:25] something called uh discrete cosine [5:28] transform. Basically using sinosoids to [5:31] to say how does that sound signal um uh [5:35] look and can I get rid of frequencies [5:37] that you can't hear and therefore [5:39] compressing the amount of information [5:40] there is. Similarly in JPEG [5:43] lossy compression you lose some of the [5:46] information but to you and I can't [5:47] really perceive it. Maybe you can but [5:49] not not always. [5:52] And then there's unknown function. We [5:53] have we have data empirical generated or [5:55] otherwise collected data that we fit [5:57] curves to all the time and uh and uh [6:00] that we do in our professional eyes. [6:04] So here's um u here's a 4 series [6:08] approximation for what's called a square [6:09] wave. So the first approximation is a [6:12] single sinosoid. But if as you add more [6:15] higher frequencies to you get you tend [6:16] to get more details and that's how you [6:18] approximate it. Can you say this is a [6:20] square wave um at the [6:24] um [6:34] um at the when when you when you add [6:36] more frequencies. Yeah, it's getting [6:37] closer, right? So so in within about [6:40] five frequencies, you're getting a good [6:42] approximation. So do you need higher [6:43] frequencies? Perhaps not. But it depends [6:45] on the use case. [6:48] In when we do MRIs, we are we are [6:51] sampling the frequency space and then [6:55] creating or approximating the image by [6:57] doing a reverse foyer transform. And um [7:00] what that what that means is that if you [7:02] just sample the lower frequencies, you [7:03] can see a little fuzzy image. It sort of [7:05] looks like what it is, but not really. [7:07] In the middle, you have something that [7:09] um looks at the edges and more more [7:12] higher uh details, but when you combine [7:14] the two, you you have the full content [7:16] of the MRI. [7:21] So this is what I mean by a known [7:23] function. When you have bunch of bunch [7:24] of points, you draw a line through it [7:26] and say, "Yeah, this sort of looks like [7:28] a line and I can approximate or predict [7:30] things based on that line." Linear [7:32] regression, something we've been doing [7:33] this problem since high school. Um so um [7:37] if I were to put that in an equation, [7:39] it's a line equation. No matter how you [7:41] slice it, it's it's basically m(x) plus [7:43] b or slope times times whatever the [7:46] variable is plus a y intercept, right? [7:51] Let's expand that to the and add a [7:53] variable to it. Instead of a line, now [7:55] you have a plane in a in a cloud of [7:57] points in a in 3D. [8:01] If you were to go into higher dimensions [8:03] or higher variables, [8:05] you I can't show that to you because you [8:07] can't really perceive it, but it becomes [8:10] um just an equation that says it's a [8:13] weight times the variable and you add [8:16] the bunch of those and then add the [8:18] constant like the y intercept. Right? [8:23] My my intention is not to teach you [8:24] math. I'm just using this as visual [8:26] tool. [8:28] So you have there's supposed to be a [8:31] slide before this. [8:34] Um [8:36] the right uh yeah I I see the right um [8:40] the right um [8:45] presentation did not load here so [8:46] apologies but it still has the [8:48] information in it. So if you if you were [8:51] to compress that equation, the long [8:52] equation that you saw in the prior [8:53] slide, it just becomes a summation of u [8:56] weight times uh the uh the original [9:00] value of each variable. [9:02] And then if you were to you can you can [9:05] um you can make it even more generalized [9:07] and say instead of the x can I say can I [9:09] a function of an x? And the answer is [9:11] yes. And in medicine uh there's there's [9:14] an equation called MEL equation that [9:16] uses the log transform regression. That [9:18] means that instead of just using X you [9:21] can use natural logarithm of of that and [9:24] that's the equation on the right side. [9:26] We use that for things um to predict [9:28] what the condition of the liver is. And [9:30] so in one of the things that I do is a [9:33] liver bypass uh called tips and this [9:36] becomes this value becomes very [9:38] important. [9:42] So the bottom line is that there are [9:44] things that we don't we estimate by [9:47] there are unknown function that we [9:48] estimate by using some of these [9:49] techniques. So how do we connect linear [9:51] regression to deep learning? [9:54] Let's first before we do that let's look [9:56] at biological neuron. So you have [9:59] dendrites [10:00] which are which are the which are the [10:02] receiving end of the information the [10:05] cell body and then the axon or the [10:07] terminal terminal axon which is the [10:09] output. [10:11] So the dendra is taking information. So [10:13] that's the input. That's the input layer [10:14] you receive. And then you have weights. [10:17] Not everything is weighted weighted the [10:19] same way, all the inputs. And then you [10:22] um you modify that signal because if you [10:24] if you have a signal that comes in and [10:27] and if you add it all up, it's going to [10:28] blow up the the neuron so that the um uh [10:31] the cell body modifies it and that's [10:33] your activation function. And then and [10:35] then it goes out. [10:40] So if you look at the multilinear [10:41] equation, it's almost identical to the [10:45] uh to the single neuron equation with [10:46] the exception of one thing which is the [10:48] activation function. So you modifying [10:50] your linear regression with the [10:52] activation function and that's truly the [10:54] power of um that introduction of the [10:56] nonlinearity is the power of of deep [10:59] learning. So each neuron is linear [11:02] regressor plus the activation function [11:04] and then you a whole bunch of neurons [11:07] and that that becomes your uh that [11:08] becomes your first layer of um [11:12] of your deep learning or the first layer [11:14] of the u of the net. [11:17] So this is um uh these are a bunch of [11:20] the first layer is red then the gray and [11:22] then additional gray layer of neurons. [11:25] But if you were to translate to a [11:26] network you have a first hidden layer, [11:27] secondhand layer and so on. So you're [11:29] taking one layer, the output of that [11:31] goes to the second layer, output of that [11:32] goes to the third layer. So it's [11:34] basically function of a function of a [11:36] function of a function. The deeper you [11:37] go, that's the deep part of deep [11:38] learning, the more complex patterns [11:41] emerge and it's basically you are just [11:43] it's cascading or um uh linear [11:46] regressors with that activation function [11:48] I'm talking about. [11:50] And there's even biological uh uh basis [11:53] for to use an activation function as [11:55] well. [11:57] So this leads us to universe [11:58] approximation theorem. What it says is [12:02] that if you have a continuous function [12:06] and on a finite domain meaning a small [12:08] amount of that uh of the whole [12:10] continuous function on a small domain [12:13] you can approximate [12:15] any function uh with arbitrary uh [12:18] arbitrary uh accuracy if you have enough [12:23] hidden layer hidden neurons in a single [12:25] layer. [12:26] So if you have enough neurons, you can [12:28] approximate any function on a small [12:30] domain. Um, and that is that is a very [12:33] powerful statement. It doesn't guarantee [12:34] anything about a how many neurons or it [12:37] doesn't say that the other architectures [12:39] will work, but at least it says that on [12:41] a certain field you you're guaranteed a [12:44] solution [12:48] and I'll give you some examples. These [12:50] are the two things I was going to talk [12:52] about due to some issues. I won't be [12:54] able to show you the deep networks, but [12:56] we can certainly um um simulation of a [12:59] deep network in in um [13:03] um [13:05] live, but I uh but I I can show you [13:08] pictures of some of the other things I [13:09] did. So, I took a signal and and said, [13:12] "Okay, this is the signal that I want to [13:14] approximate. This is the known function [13:15] I want to approximate. And I'm going to [13:17] use the orange dots as my as my test set [13:22] and the blue dots as the u as the [13:24] training set. [13:28] In the in the first first time around I [13:30] said okay let's use 30 neurons in a [13:32] single layer. And it's a terrible [13:34] approximation as you can see. [13:40] Then I took um uh then I said okay let's [13:43] increase that by a couple orders of [13:44] magnitude. So now 5,000 neurons you get [13:47] the first sort of approximation the [13:49] shape sort of looks like it but it [13:50] doesn't have the the details. [13:53] So um so that's a that's a problem. When [13:56] I increase and I changed the [13:58] architecture and added more hidden [14:00] layers it requires less neurons. It [14:02] still doesn't look any anywhere close to [14:04] what we what we want. But then when I [14:06] add more neurons [14:08] um [14:10] to 200, 500 and 200 in the three layers, [14:13] it sort of resembles what we started [14:15] with. [14:17] When I add even more, [14:20] it gets um it gets better, but not that [14:22] much better. [14:24] And then when I add even more and an [14:27] additional layer, you see that some of [14:29] the edge effects [14:31] um get better. the edges are better [14:34] approximated than prior one. Um and uh [14:36] so after a while the more neurons you [14:40] add you may not there might be a [14:41] diminishing return. [14:46] So this is uh this is what um I would [14:49] recommend that you go to. I cannot show [14:51] you live because I don't have a keyboard [14:53] because I was going to show you to [14:54] adding some of those neurons. So I'll [14:56] explain to you um there are there are a [14:58] bunch of um a bunch of functions on it. [15:01] One of them is a is a two uh two layers [15:05] of what looks like a heart and you want [15:07] to separate the data points. When you [15:09] use a when you use an activation [15:11] function that's linear uh it doesn't [15:14] really work. It's uh it gets to a [15:16] certain point and just stays there. Um [15:18] and when you use more advanced functions [15:20] like u like rectal linear unit or [15:23] hyperbolic tan u then the nonlinearity [15:26] comes into play and it does a much [15:28] better job approximating a function. And [15:30] that's the power. So when I said that [15:32] equation that the activation function [15:34] really matters, one of the things the [15:37] newest approximation theorem says as [15:38] long as the activation function is [15:40] nonlinear then you can do that. If it is [15:42] not then you lose the power of of deep [15:45] learning. So it's super critical in my [15:47] mind that we um uh we have that um [15:50] nonlinear uh function in because that's [15:53] what uh that's what the math says. [15:57] uh uh this this particular site deeper [16:00] learning it's a it's a TensorFlow um um [16:03] uh uh simulator and I really encourage [16:06] you to go around fool around with it [16:07] because it really gives you the [16:08] intuitive understanding that you're [16:09] looking for. [16:12] So in summary uh this is again this is [16:14] an old slide so I'll just uh I'll go off [16:17] um and and whoever wants it I can send [16:19] you the presentation. [16:21] So uh number one deep learning is has [16:24] mathematical foundation and can be [16:26] thought of as an approximation of [16:28] unknown function. [16:30] So this is the controversial statement I [16:32] was going to make. We've all been taught [16:34] that deep learning is not transparent. [16:36] If you use the definition we talked [16:38] about earlier, it is transparent. We [16:40] know the math. We know how we're [16:41] training it. We know um uh we know that [16:44] um um we know the inner workings of it [16:48] once once we understand uh the structure [16:50] of it. What we don't know is what we [16:54] can't do is um explain what's inside it. [16:57] So it's not it may not be explainable [16:59] depending on the complexity of the [17:00] network. It may not be interpretable [17:03] because if it's too much information [17:05] then you may not be able to do that. [17:06] There are techniques like um um where [17:09] you can have we can you can back [17:11] propagate back and do create heat maps [17:13] that that tells you what are the inputs [17:15] that are driving the decision. So there [17:17] are some techniques that can do that but [17:19] it is not as as interpretable or um um [17:24] as let's say linear regression itself [17:27] right as if single equation linear [17:28] regression is pretty pretty easy to [17:30] explain because the the u the [17:32] coefficients tell you what what what's [17:34] driving the uh the output. But when you [17:36] have many of those regressors and deep [17:39] it becomes much more challenging but it [17:41] is not impossible. It does give you we [17:43] can peak into it. [17:50] So I hope this gives you some [17:51] understanding of connection between um [17:54] between linear regression and a neuron [17:56] and a neuron to deeper um deeper [17:58] networks. Thank you. [18:04] Let's sew some love to Mr. Patel. Thank [18:06] you so much again.