Air Pollution: A Global Health Crisis

• 7 million premature deaths annually (WHO)
• More than malaria, HIV, and road accidents combined
• 91% of deaths in low/middle-income countries

Current Air Quality Monitoring in India

Metric	Value
CPCB stations	~600
Population	1.4 billion
People per station	2.3 million
Area per station	5,400 km²

Takeaway: Severely under-monitored for a country of this size and pollution levels

Why Not Just Add More Sensors?

Full CPCB CAAQMS (BAM + gases + shelter): ₹2-3 crore+ per station

• Large-scale deployment → extremely costly

Current CPCB network: ~600 stations

Smart placement is critical — every sensor must maximize value!

The Core Question

Given a limited budget of new sensors:

Where should we place them?

Goal: Maximize information about the entire region

Problem Formulation

— Existing sensors

— Candidate locations ( points)

— Target region to predict

— Budget for new sensors

Goal: Select from to maximize info about

Optimal Sensor Placement (OSP)

1. Surrogate Model — Predicts values + uncertainty; must be differentiable
2. Acquisition Function — Scores candidates; guides placement

Acquisition 1: Random

Pick locations uniformly at random from candidates

In plain English:

"Close your eyes and point at the map times"

• No optimization — pure luck
• Ignores data completely
• Baseline for comparison

Acquisition 2: Maximum Variance (MaxVar)

Greedily pick the location with highest uncertainty

In plain English:

"Where are we most uncertain? Put a sensor there."

Repeat times, each time adding the selected sensor to context.

Pros: Fast , intuitive

Cons: Doesn't maximize information gain with minimum sensors

The MI Objective: What We Want to Maximize

Mutual Information: How much does knowing reduce uncertainty about ?

Reading this: "Given existing sensors , how much does adding new sensors reduce our uncertainty about the target region?"

MI vs MaxVar: What's the Difference?

For Gaussian outputs (GP/Neural Process): → both use variance

• MaxVar: "where am I most uncertain?"
• MI: "what reduces uncertainty everywhere?"

Problem: Exact MI requires searching subsets — combinatorially explosive!

Acquisition 3: Greedy MI (Standard Approach)

Since exact MI is intractable, use greedy approximation: select one sensor at a time

Algorithm: For each of rounds, evaluate all candidates → pick best

Complexity: evaluations

For India (): computationally infeasible!

Our Solution: GD-MI

Key insight: Don't search discrete candidates — optimize coordinates directly!

	Greedy MI	GD-MI (Ours)
Search space	Discrete ( candidates)	Continuous (coordinates)
Optimization	Enumerate all	Gradient descent
Complexity		iterations
Scalability		✓ Any

GD-MI: How It Works

What we optimize: sensor locations, each a (lat, lon) pair

Objective: Minimize average variance over target region

where — predicted values at proposed locations

Algorithm: Initialize → Forward (get ) → Backward () → Update → Repeat

Key: Model is frozen — only the coordinates are optimized

Surrogate Model: Why Neural Processes?

GD-MI needs a model that provides:

TNP-D (Transformer Neural Process) satisfies all!

Surrogate Model: Transformer Neural Process (TNP-D)

• Predicts PM and uncertainty
• Fully differentiable
• Fast parallel inference

Model	RMSE ↓	NLL ↓
Gaussian Process	5.16	-0.19
Convolutional GNP	5.31	-0.30
Transformer NP	4.90	-0.44

Experiment 1: Regional Validation

Madhya Pradesh — a central Indian state (~size of Germany, =308 candidates)

Results (k=9 sensors):

Method	RMSE ↓
Random	7.2
MaxVar	6.3
GD-MI	5.8
Greedy MI	5.6

GD-MI within 0.2 RMSE of Greedy MI (gold standard)

Experiment 2: India-Scale (The Real Test)

Greedy MI infeasible () — GD-MI shines here

Key findings:

• GD-MI 4% better than MaxVar
• Gap grows with more sensors
• Consistent across budgets

Why Does GD-MI Win? Qualitative Analysis

MaxVar (blue dots)

• Optimizes for single-point uncertainty
• Doesn't consider target region

GD-MI (red dots)

• Optimizes for entire target region
• Maximizes information gain

MI objective guides placement for maximum coverage

Scalability: GD-MI vs Greedy MI

• Greedy MI: runtime grows with pool size
• GD-MI: constant time regardless of
• At : GD-MI is ~100× faster

GD-MI: Limitations

• Optimizes in continuous space
  → cannot enforce real-world deployment constraints

• May select invalid / impractical locations
  (e.g., inaccessible terrain)

• Assumes uniform importance
  → ignores population / exposure differences

• Non-convex optimization
  → solutions depend on initialization

Regional Budget Constraints

From Constraints to Selection

Each sensor must choose one location from its state's candidate pool

From Constraints to Selection

Represent this as a probability distribution over candidates

From Constraints to Selection

argmax picks the highest probability location

From Constraints to Selection

But argmax is not differentiable — gradients cannot flow through it

Gumbel-Softmax: Making Sampling Differentiable

GSM: How It Works

Don't worry about the complexity — let's go term by term

GSM: How It Works

— learnable logits, one row per sensor, one column per candidate location

The learnable logit matrix W

GSM: How It Works

— regional mask: sets out-of-state entries to → probability = 0

Enforcing regional budgets via masking

GSM: How It Works

— Gumbel noise: constant in backward pass — this is the reparameterisation trick

GSM: How It Works

— temperature: high → explore broadly, low → commit to one location

GSM: How It Works

• — learnable logits
• — regional mask
• — Gumbel noise
• — temperature

Does It Converge? Training Dynamics

Regional Validation: GSM vs Baselines

Continental Results: Unconstrained Deployment

Main Result: Constrained Deployment

Qualitative Comparison of Deployments

GSM: Limitations

• Uses a continuous relaxation
  → introduces approximation error

• Sensitive to temperature (τ) scheduling
  → affects convergence stability

• Non-convex optimization over logits
  → sensitive to initialization; multiple local optima

Future Work

• Equity-aware placement
  → weight by population, pollution, vulnerability

• Dynamic sensing
  → mobile sensors, adaptive placement

• Multi-pollutant optimization
  → PM₂.₅, NO₂, O₃ jointly

• Improved discrete relaxations
  → Top-K, optimal transport (Sinkhorn)

Publications

• Scalable Air-Quality Sensor Placement via Gradient-Based Mutual Information Maximization (AAAI 2026 — AI for Social Impact)
Zeel B Patel, Vinayak Rana, Nipun Batra

• Large-Scale Air-Quality Sensor Placement via Joint Optimization under Regional Constraints (Under Review)
Vinayak Rana, Neerja Kasture, Anura Mantri, Nipun Batra