Dimensionally-Constrained CNN to TopOC Transfer Learning

We are working on transfer learning that combines convolutional neural networks (CNNs) with Topological Object Classification (TopOC) methods. This memo outlines our design approach to dimensionally-constrained models that maintain representational integrity through the transfer learning process, while targeting deployment to resource-constrained hardware through our Fidelity Framework compilation pipeline.

The design uses our Clef language’s Units of Measure (UMX) system to enforce dimensional constraints across the model architecture. Holding those constraints at compile time gives the models more regular computational patterns, which the framework can lower toward FPGAs and custom ASICs. Compile-time dimensional checking paired with direct hardware lowering is the combination we are pursuing for edge environments where conventional frameworks carry their shape checks into runtime.

The Dimensional Integrity Challenge in Transfer Learning

Transfer learning between CNNs and topological representations introduces challenges that conventional frameworks address only at runtime:

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 trial-and-error experimentation, which leaves the resulting models harder to deploy. Our approach moves these dimensional constraints to compile time, where the type checker resolves them once.

Clef’s Units of Measure: A Foundation for Dimensional Safety

Our architecture uses Clef’s type system to enforce dimensional constraints across the 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 Clef 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>) =
    // Channel -> Channel * 2
    let conv1 = Conv2D<Channel, Channel * 2>(
        kernelSize = (3<Height>, 3<Width>),
        stride = (1<Height>, 1<Width>),
        padding = PaddingType.Same)

    // dimension-preserving pool
    let pool1 = MaxPool2D<Channel * 2>(
        kernelSize = (2<Height>, 2<Width>),
        stride = (2<Height>, 2<Width>))

    // ...

    DimensionalSequential [conv1; pool1; ...]

Each layer’s input and output dimensions are checked at compile time, so the feature extraction process maintains dimensional integrity across the stack. This removes the class of shape-mismatch errors that conventional implementations catch only at runtime.

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 =

    // Feature -> PersistenceDegree
    let persistenceDiagrams =
        computePersistentHomology<Feature, PersistenceDegree> features

    // PersistenceDegree -> HomologyClass
    let betti =
        extractBettiNumbers<PersistenceDegree, HomologyClass> persistenceDiagrams

    betti

This transformation holds its dimensional constraints while converting between two different representation spaces, a conversion that conventional frameworks leave unchecked until runtime.

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> =

    let landscapes =
        computePersistenceLandscapes<PersistenceDegree, HomologyClass> topo

    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

In the compilation pathway we are building, the dimensional information from Clef’s UMX system gives the lowering passes more to work with when targeting 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 compilation pipeline we envision draws on the dimensional information in several ways:

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

Carrying the dimensional constraints through to the target removes the translation layers and runtime shape checks that conventional toolchains spend resources on in accelerated environments.

Practical Performance Analysis

Our design anticipates measurable advantages over conventional approaches:

We attribute these projected gains to the dimensional constraints, which hold the correct relationships across the model architecture and let the compiler specialize against them.

Representational Integrity Across the Pipeline

Our approach holds representational integrity across the 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.

Tracing dimensional relationships through the stack gives the structural properties to the verifier directly. Where a black-box model offers only empirical testing, the dimensional invariants here are properties the type system establishes from the types themselves.

ASIC Integration Through MLIR Lowering

For custom ASIC deployment, our Fidelity framework is designed 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 is meant to put CNN-to-TopOC models on devices that resource constraints would otherwise rule out.

Dimensionally-Grounded Inertial Navigation System: A Reference Design

We have outlined specifications for an Inertial Navigation System (INS) that applies our dimensionally-constrained approach, aiming for high precision with modest computational requirements. The reference design shows how our CNN-to-TopOC architecture with UMX dimensional safety carries into safety-critical applications that must run on resource-constrained hardware.

Physical Dimensions as First-Class Types

The INS design begins with a 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
    ]

Topological continuity constraints enforce temporal and spatial consistency in the navigation solution, which is where drift accumulates 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 is meant to use the dimensional constraints to produce FPGA implementations that hold computational precision while keeping resource usage low.

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

The dimensional discipline also points toward safety certification, because the properties a certifier asks about line up with the framework’s tiered verification rather than with a test campaign. The structural ones, dimensional consistency and the grade of the navigation state, follow from the types themselves at no cost. The harder ones, like a bounded-acceleration limit or the topological consistency of the trajectory, are obligations we discharge further up the proof tiers, by the solver where they are decidable and by the relational layer where they are not. Each property travels with its evidence at the tier its difficulty demands, rather than as a claim that testing can only sample.

The Path Forward: Dimensions as Formal Verification

Beyond our INS reference design, our broader aim is to carry these constraints from shape correctness into deeper mathematical properties, for instance that small perturbations of an input preserve a model’s topological features. A property of that kind is not a free theorem from the types; it is a higher-tier obligation, the sort the proof stack reaches where structural typing alone cannot, through the solver’s probabilistic fragment or the relational layer that reasons about a model’s behavior across paired runs.

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.

Where the Design Stands

Our dimensionally-constrained CNN to TopOC transfer learning approach carries Clef’s Units of Measure system through the model architecture and into the compilation pipeline. The constraints that conventional frameworks check at runtime are resolved here at compile time, which is what brings these models within reach of resource-constrained targets.

We have found no other representative implementation of this combination, dimensional typing across a CNN-to-TopOC transfer and direct lowering to FPGA and ASIC targets, in the standing literature we have reviewed. From edge devices to custom ASICs, the pathway we are building lets the same dimensional discipline travel from the model definition down to the hardware description.

The work ahead is to extend this approach to other transfer learning domains and to push the dimensional constraint system further up our verification tiers, from the shape correctness the types already carry toward the deeper properties the higher tiers reach. That is where our interest lies as the work continues.