At SpeakEZ, we are working on transformative approaches to transfer learning that combine convolutional neural networks (CNNs) with Topological Object Classification (TopOC) methods. This memo outlines our design approach to creating dimensionally-constrained models that maintain representational integrity throughout the transfer learning process while enabling deployment to resource-constrained hardware through our Fidelity Framework compilation pipeline.

By leveraging F#’s Units of Measure (UMX) system to enforce dimensional constraints across the entire model architecture, we achieve not only safer and more reliable models but also significantly more efficient computational patterns that can be directly compiled to FPGAs and custom ASICs. This combination of type safety and direct hardware compilation creates a unique advantage for deploying sophisticated AI capabilities in edge environments where traditional approaches fall short.

The Dimensional Integrity Challenge in Transfer Learning

Transfer learning between CNNs and topological representations introduces fundamental challenges that conventional frameworks struggle to address:

1. Dimensional Consistency: CNNs operate on grid-structured data with specific channel, height, and width dimensions, while topological representations focus on abstract structural invariants

2. Feature Space Mapping: Translating between convolutional feature spaces and topological structures requires maintaining precise dimensional relationships

3. Compilation Target Constraints: Hardware accelerators like FPGAs and ASICs impose strict requirements on memory layout and computational patterns

Conventional approaches using Python-based frameworks handle these challenges through runtime checks and extensive trial-and-error experimentation, leading to inefficient models and deployment challenges. Our approach fundamentally reimagines this process through statically-verified dimensional constraints.

F#’s Units of Measure: A Foundation for Dimensional Safety

Our architecture leverages F#’s sophisticated type system to enforce dimensional constraints throughout the entire model pipeline:

module ModelDimensions =
    // Basic dimension types
    type [<Measure>] Channel
    type [<Measure>] Height
    type [<Measure>] Width
    type [<Measure>] Feature
    type [<Measure>] BatchSize
    
    // Topological dimension types
    type [<Measure>] PersistenceDegree
    type [<Measure>] HomologyClass
    type [<Measure>] BettiNumber
    
    // Combined dimensions for tensors
    type ImageTensor = Tensor<Channel * Height * Width>
    type FeatureTensor = Tensor<Channel * Feature>
    type TopologicalTensor = Tensor<PersistenceDegree * HomologyClass>
    
    // For shape tracking in convolutional operations
    type ConvKernel<[<Measure>] 'in, [<Measure>] 'out> = 
        Tensor<'in * 'out * Height * Width>

This type system provides compile-time guarantees that operations maintain dimensional consistency without any runtime overhead. Unlike Python-based approaches that must repeatedly validate tensor shapes during execution, our F# implementation resolves these concerns entirely at compile time.

CNN to TopOC Transfer Architecture

Our CNN to TopOC transfer learning architecture consists of three main components, each leveraging dimensional constraints:

1. Dimensionally-Constrained Backbone CNN

The CNN backbone extracts features with strictly enforced dimensional relationships:

let createBackbone (inputShape: int<Channel> * int<Height> * int<Width>) =
    // Type-safe convolution with dimensional guarantees
    let conv1 = Conv2D<Channel, Channel * 2>(
        kernelSize = (3<Height>, 3<Width>),
        stride = (1<Height>, 1<Width>),
        padding = PaddingType.Same)
    
    // Dimension-preserving pooling
    let pool1 = MaxPool2D<Channel * 2>(
        kernelSize = (2<Height>, 2<Width>),
        stride = (2<Height>, 2<Width>))
    
    // Subsequent layers maintain dimensional relationships
    // ...
    
    // Returns dimensionally-typed feature tensors
    DimensionalSequential [conv1; pool1; ...]

Each layer’s input and output dimensions are verified at compile time, ensuring that the feature extraction process maintains dimensional integrity throughout. This eliminates an entire class of errors that plague traditional implementations.

2. Topological Feature Transformation

The critical bridge between CNN features and topological structures is implemented with dimensional safety:

// Transform CNN features to topological representation
let cnnToTopological 
    (features: FeatureTensor) 
    : TopologicalTensor =
    
    // Compute persistent homology with dimension tracking
    let persistenceDiagrams = 
        computePersistentHomology<Feature, PersistenceDegree> features
    
    // Extract Betti numbers with dimension preservation
    let betti = 
        extractBettiNumbers<PersistenceDegree, HomologyClass> persistenceDiagrams
    
    // Return topologically structured tensor with verified dimensions
    betti

This transformation maintains dimensional safety constraints while converting between fundamentally different representation spaces - a challenging task that typically introduces errors in conventional frameworks.

3. TopOC Classification Head

The topological classification component leverages the structural invariants with dimensional integrity:

// Classification head using topological features
let topologicalClassifier 
    (topo: TopologicalTensor) 
    (numClasses: int) 
    : Tensor<int> =
    
    // Dimension-preserving persistence landscape computation
    let landscapes = 
        computePersistenceLandscapes<PersistenceDegree, HomologyClass> topo
    
    // Final classification while maintaining dimensional constraints
    let logits = 
        fullyConnected<HomologyClass * PersistenceDegree, int> landscapes numClasses
    
    logits

By maintaining dimensional constraints throughout this process, we ensure that the topological invariants are correctly preserved and utilized in the final classification decision.

Direct Hardware Compilation Through Fidelity Framework

The dimensional safety provided by F#’s UMX system creates unique advantages when compiling these models to hardware accelerators through our Fidelity Framework:

// Configure model for FPGA deployment
let fpgaConfig = 
    PlatformConfig.compose 
        [withPlatform PlatformType.FPGA;
         withMemoryModel MemoryModelType.Constrained;
         withBitPrecision PrecisionType.FP16;
         withVectorization VectorizationType.Spatial]
        PlatformConfig.base'

// Compile model directly to FPGA bitstream
let compiledModel = 
    FidelityCompiler.compile
        model
        fpgaConfig
        optimizationLevel = OptimizationLevel.Aggressive

The Fidelity compilation pipeline leverages several critical advantages:

1. Zero-Copy Memory Layout: The dimensional constraints allow for precise memory alignment and zero-copy operations specific to the target hardware

2. Dimension-Aware Optimization: The compiler can make aggressive optimizations based on the known dimensional relationships established through UMX

3. Direct MLIR Lowering: The dimensionally-constrained operations map directly to efficient MLIR patterns for FPGA and ASIC targets

4. Binary Precision Adaptation: Models can be automatically adapted to binary or reduced-precision representations while maintaining dimensional integrity

This approach transforms how models are deployed to custom hardware, eliminating the translation layers and runtime checks that typically consume resources in accelerated environments.

Practical Performance Analysis

Our design anticipates measurable advantages over conventional approaches:

The dimensional constraints provide both computational efficiency and accuracy improvements by enforcing correct relationships throughout the model architecture.

Representational Integrity Across the Pipeline

One of the most powerful aspects of our approach is the maintenance of representational integrity across the entire pipeline, from model definition to hardware deployment:

// Define an end-to-end dimensionally-constrained pipeline
let pipeline = Pipeline.create()
    |> Pipeline.add (ImagePreprocessor<Height, Width, Channel>())
    |> Pipeline.add backbone
    |> Pipeline.add topologicalTransformer
    |> Pipeline.add classificationHead
    |> Pipeline.compile fpgaConfig

This unified pipeline maintains dimensional constraints from raw input through to final classification, with the Fidelity Framework’s compilation toolchain preserving these relationships through to the hardware implementation.

The ability to trace dimensional relationships through the entire stack creates unprecedented opportunities for formal verification and safety guarantees. Unlike black-box models that can only be empirically tested, our approach allows for mathematical proofs of dimensional invariants.

ASIC Integration Through MLIR Lowering

For custom ASIC deployment, the Fidelity framework has designs to provide direct pathways from dimensionally-constrained models to hardware descriptions:

// ASIC configuration with dimensional awareness
let asicConfig =
    PlatformConfig.compose
        [withPlatform PlatformType.ASIC;
         withSiliconTarget SiliconTarget.TSMC7nm;
         withClockDomain ClockDomain.Single;
         withPowerOptimization PowerOptimizationType.DynamicScaling]
        PlatformConfig.base'

// Direct lowering to HDL through MLIR
let asicImplementation =
    FidelityCompiler.lowerToHDL
        model
        asicConfig
        hdlFormat = HDLFormat.SystemVerilog

The dimensional constraints allow the compiler to generate optimal circuit designs by:

1. Eliminating Redundant Dimension Checks: Traditional implementations must verify tensor dimensions at runtime, requiring additional circuitry

2. Optimizing Memory Access Patterns: Knowing exact dimensions enables optimal memory layout specific to the silicon implementation

3. Parallelizing Across Dimensional Units: Operations across independent dimensions can be safely parallelized without verification overhead

4. Specialized Circuit Generation: Custom circuits for topological operations can be generated based on the precise dimensional constraints

This approach allows deployment of sophisticated CNN-to-TopOC models in environments that would traditionally be inaccessible due to resource constraints.

Dimensionally-Grounded Inertial Navigation System: A Reference Design

At SpeakEZ, we have outlined specifications for an Inertial Navigation System (INS) that leverages our dimensionally-constrained approach to achieve high precision with minimal computational requirements. This reference design demonstrates how our CNN-to-TopOC architecture with UMX dimensional safety can be applied to safety-critical applications requiring deployment on resource-constrained hardware.

Physical Dimensions as First-Class Types

The INS design begins with a comprehensive type system representing physical quantities:

module PhysicalDimensions =
    // Basic physical units
    type [<Measure>] m      // meter
    type [<Measure>] s      // second
    type [<Measure>] kg     // kilogram
    type [<Measure>] rad    // radian
    
    // Derived units for inertial navigation
    type [<Measure>] mps = m/s               // velocity
    type [<Measure>] mps2 = m/s^2            // acceleration
    type [<Measure>] radps = rad/s           // angular velocity
    type [<Measure>] radps2 = rad/s^2        // angular acceleration
    type [<Measure>] T                       // tesla (magnetic field)
    
    // Sensor-specific tensor types
    type AccelerometerTensor = Tensor<mps2 * Channel * Time>
    type GyroscopeTensor = Tensor<radps * Channel * Time>
    type MagnetometerTensor = Tensor<T * Channel * Time>
    
    // Navigation state types
    type PositionVector = Vector<m * Dimension>
    type VelocityVector = Vector<mps * Dimension>
    type OrientationQuaternion = Quaternion<rad>

This type system ensures that physical laws are respected throughout all computations, preventing errors like adding velocity to position without integration over time.

Sensor Fusion with Dimensional Integrity

The core of the INS design is a sensor fusion pipeline that maintains dimensional correctness:

// CNN-based sensor fusion with topological continuity constraints
let createInertialNavigationSystem() =
    // Sensor-specific feature extractors with dimensional constraints
    let accelNet = 
        CNNModel<AccelerometerTensor, FeatureTensor>()
        |> Conv1D<Channel, Feature>(kernelSize = 5<Time>, stride = 1<Time>)
        |> InstanceNorm1D<Feature>()
        |> ReLU()
        |> MaxPool1D<Feature>(kernelSize = 2<Time>, stride = 2<Time>)
        // Additional layers...
    
    let gyroNet = 
        CNNModel<GyroscopeTensor, FeatureTensor>()
        // Similar architecture with gyroscope-specific parameters
    
    let magNet = 
        CNNModel<MagnetometerTensor, FeatureTensor>()
        // Similar architecture with magnetometer-specific parameters
    
    // Topological continuity enforcement layer
    let topologicalConstraintLayer = 
        TopOCFusion<FeatureTensor * 3, TopologicalTensor>()
        |> PersistenceLayer<Feature, PersistenceDegree>()
        |> HomologyTrackingLayer<PersistenceDegree, HomologyClass>()
    
    // Final navigation state estimator with physical constraints
    let stateEstimator =
        TopologicalTensor
        |> DimensionalDense<PersistenceDegree * HomologyClass, m * Dimension>()
        |> PhysicalConstraintLayer<m * Dimension>()
    
    // Complete fusion pipeline
    DimensionalSequential [
        ParallelProcessing [accelNet; gyroNet; magNet]
        topologicalConstraintLayer
        stateEstimator
    ]

The key innovation here is the use of topological continuity constraints that enforce temporal and spatial consistency in the navigation solution, addressing the fundamental challenge of drift in inertial systems.

FPGA Implementation Specifications

We have outlined specifications for deploying this INS design on FPGA hardware:

// FPGA-specific optimizations for the INS
let configureForFPGA (insModel: Model<SensorData, NavigationState>) =
    // Configure for Xilinx Ultrascale+ target
    let fpgaConfig = 
        PlatformConfig.compose
            [withPlatform PlatformType.FPGA;
             withFPGAVendor FPGAVendor.Xilinx;
             withFamily "Ultrascale+";
             withClockRate 200<MHz>;
             withMemoryInterface MemoryInterface.DDR4;
             withPrecision PrecisionType.Mixed]
            PlatformConfig.base'
    
    // Specific optimizations for INS workloads
    let insOptimizations =
        OptimizationConfig.compose
            [withPipelining PipeliningLevel.Aggressive;
             withResourceSharing ResourceSharingLevel.Balanced;
             withOperatorBalancing true;
             withDSPUtilization DSPUtilizationLevel.Maximum]
            OptimizationConfig.default'
    
    // Generate optimized bitstream
    FidelityCompiler.compileForFPGA
        insModel
        fpgaConfig
        insOptimizations

This configuration leverages the dimensional constraints to create highly optimized FPGA implementations that maintain computational precision while minimizing resource usage.

Performance and Validation

Our reference design specifications indicate the following projected performance characteristics:

The improvement in position drift comes from the topological continuity constraints that prevent physically impossible state transitions, while the efficiency gains enable deployment on smaller, lower-power FPGA devices suitable for mobile and autonomous platforms.

Safety Certification Pathway

Perhaps most significantly, our dimensionally-constrained approach creates a clear pathway toward safety certification:

// Safety property specification for INS
let specifyINSSafetyProperties (ins: InertialNavigationSystem) =
    [
        // Physical continuity of position
        Property.continuous<m * Dimension> ins.Position;
        
        // Bounded acceleration (physical limitation)
        Property.bounded<mps2> ins.Acceleration (-20.0<mps2>, 20.0<mps2>);
        
        // Conservation of energy within sensor noise bounds
        Property.conserves ins.TotalEnergy EnergyTolerance.SensorNoise;
        
        // Topological consistency of trajectory
        Property.topologicalInvariant ins.Trajectory HomologyClass.PathConnected
    ]

These properties can be formally verified through the F*/Fidelity bridge, moving safety-critical navigation systems from traditional test-based validation to provable correctness guarantees.

The Path Forward: Dimensions as Formal Verification

Beyond our INS reference design, we are exploring broader applications of dimensional constraints for formal verification:

// Formal verification of topological invariance
let verifyTopologicalInvariance (model: Model<'input, 'output>) =
    // Prove that small perturbations in input preserve 
    // topological features in output
    let theorem = 
        forall (x: 'input) (delta: 'input) ->
            when (norm delta < epsilon) ->
                topologicalDistance(model.forward(x), 
                                   model.forward(x + delta)) < delta'
    
    // Verify through F*/Fidelity bridge
    FidelityVerifier.prove theorem

While these capabilities are still being developed, they represent the logical extension of our dimensional constraint system - moving from ensuring correct shapes to proving deeper mathematical properties of the models we deploy.

Conclusion: A New Standard for Model Safety and Efficiency

At SpeakEZ, our dimensionally-constrained CNN to TopOC transfer learning approach represents a fundamental advance in both model safety and deployment efficiency. By leveraging F#’s Units of Measure system throughout the model architecture and compilation pipeline, we’ve created a comprehensive solution to challenges that have limited the deployment of sophisticated AI in resource-constrained environments.

The Fidelity Framework’s direct compilation pathway from dimensionally-constrained models to custom hardware opens new possibilities for AI deployment across the computing spectrum. From edge devices to custom ASICs, our approach enables a level of efficiency and safety that simply isn’t possible with conventional frameworks.

Our next steps include expanding this approach to other transfer learning domains and further enhancing the formal verification capabilities of our dimensional constraint system. As the industry continues to push toward more efficient and reliable AI systems, we believe this approach represents the future of model design and deployment.